Missing gcc, build-essential

Ubuntu has already ended the support for its 14.04 release (except for ESM which is a support extension for servers that has to be paid for).
You should upgrade to a supported Ubuntu release.

And for your installation problem, what output do you receive for the following commands (to be executed in a terminal window):

uname -a
lsb_release -crid
apt-cache policy gcc build-essential
sudo apt-get install gcc build-essential

Hi ! Manfred Hampl, I provide info here.

====================================

mkido@mkido-AO531h:~$ uname -a
Linux mkido-AO531h 3.13.0-24-generic #46-Ubuntu SMP Thu Apr 10 19:08:14 UTC 2014 i686 i686 i686 GNU/Linux

mkido@mkido-AO531h:~$ lsb_release -crid
Distributor ID: Ubuntu
Description: Ubuntu 14.04 LTS
Release: 14.04
Codename: trusty

mkido@mkido-AO531h:~$ apt-cache policy gcc build-essential
gcc:
  Installed: 4:4.8.2-1ubuntu6
  Candidate: 4:4.8.2-1ubuntu6
  Version table:
 *** 4:4.8.2-1ubuntu6 0
        500 http://us.archive.ubuntu.com/ubuntu/ trusty/main i386 Packages
        100 /var/lib/dpkg/status
build-essential:
  Installed: (none)
  Candidate: 11.6ubuntu6
  Version table:
     11.6ubuntu6 0
        500 http://us.archive.ubuntu.com/ubuntu/ trusty/main i386 Packages

mkido@mkido-AO531h:~$ sudo apt-get install gcc build-essential
[sudo] password for mkido:
Reading package lists... Done
Building dependency tree
Reading state information... Done
gcc is already the newest version.
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies:
 build-essential : Depends: g++ (>= 4:4.4.3) but it is not going to be installed
E: Unable to correct problems, you have held broken packages.
mkido@mkido-AO531h:~$

THANKS

What is now the output of

apt-cache policy g++
sudo apt-get install g++ build-essential

Hi ! There, thanks, here it is,

mkido@mkido-AO531h:~$ apt-cache policy g++
g++:
  Installed: (none)
  Candidate: 4:4.8.2-1ubuntu6
  Version table:
     4:4.8.2-1ubuntu6 0
        500 http://us.archive.ubuntu.com/ubuntu/ trusty/main i386 Packages

mkido@mkido-AO531h:~$ sudo apt-get install g++ build-essential
[sudo] password for mkido:
Reading package lists... Done
Building dependency tree
Reading state information... Done
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies:
 g++ : Depends: g++-4.8 (>= 4.8.2-5~) but it is not going to be installed
E: Unable to correct problems, you have held broken packages.
mkido@mkido-AO531h:~$

END

What is now the output of

apt-cache policy g++-4.8
sudo apt-get install g++-4.8 build-essential

Hi ! There, sorry for the slow reply,

mkido@mkido-AO531h:~$ apt-cache policy g++-4.8
g++-4.8:
  Installed: (none)
  Candidate: 4.8.2-19ubuntu1
  Version table:
     4.8.2-19ubuntu1 0
        500 http://us.archive.ubuntu.com/ubuntu/ trusty/main i386 Packages

mkido@mkido-AO531h:~$ sudo apt-get install g++-4.8 build-essential
[sudo] password for mkido:
Reading package lists... Done
Building dependency tree
Reading state information... Done
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies:
 g++-4.8 : Depends: gcc-4.8-base (= 4.8.2-19ubuntu1) but 4.8.4-2ubuntu1~14.04.4 is to be installed
           Depends: gcc-4.8 (= 4.8.2-19ubuntu1) but 4.8.4-2ubuntu1~14.04.4 is to be installed
           Depends: libstdc++-4.8-dev (= 4.8.2-19ubuntu1) but it is not going to be installed
E: Unable to correct problems, you have held broken packages.
mkido@mkido-AO531h:~$

END

What output do you receive for the commands

sudo apt-get update
apt-cache policy g++-4.8 gcc-4.8

Hi ! There, here they are.

2021-09-04.hplip_continuation.txt

mkido@mkido-AO531h:~$ sudo apt-get update
[sudo] password for mkido:
Get:1 http://dl.google.com stable InRelease [1,811 B]
Ign http://dl.google.com stable InRelease
Ign http://us.archive.ubuntu.com trusty InRelease
Ign http://extras.ubuntu.com trusty InRelease
Hit http://us.archive.ubuntu.com trusty Release.gpg
Hit http://us.archive.ubuntu.com trusty Release
Get:2 http://extras.ubuntu.com trusty Release.gpg [72 B]
Hit http://us.archive.ubuntu.com trusty/main Sources
Hit http://extras.ubuntu.com trusty Release
Hit http://us.archive.ubuntu.com trusty/restricted Sources
Hit http://extras.ubuntu.com trusty/main Sources
Hit http://us.archive.ubuntu.com trusty/universe Sources
Hit http://extras.ubuntu.com trusty/main i386 Packages
Hit http://us.archive.ubuntu.com trusty/multiverse Sources
Hit http://us.archive.ubuntu.com trusty/main i386 Packages
Hit http://us.archive.ubuntu.com trusty/restricted i386 Packages
Err http://dl.google.com stable/main i386 Packages
  404 Not Found [IP: 2404:6800:4004:813::200e 80]
Hit http://us.archive.ubuntu.com trusty/universe i386 Packages
Ign http://dl.google.com stable/main Translation-en_US
Hit http://us.archive.ubuntu.com trusty/multiverse i386 Packages
Ign http://dl.google.com stable/main Translation-en
Hit http://us.archive.ubuntu.com trusty/main Translation-en
Hit http://us.archive.ubuntu.com trusty/multiverse Translation-en
Hit http://us.archive.ubuntu.com trusty/restricted Translation-en
Hit http://us.archive.ubuntu.com trusty/universe Translation-en
Ign http://extras.ubuntu.com trusty/main Translation-en_US
Ign http://extras.ubuntu.com trusty/main Translation-en
Ign http://us.archive.ubuntu.com trusty/main Translation-en_US
Ign http://us.archive.ubuntu.com trusty/multiverse Translation-en_US
Ign http://us.archive.ubuntu.com trusty/restricted Translation-en_US
Ign http://us.archive.ubuntu.com trusty/universe Translation-en_US
Fetched 1,883 B in 9s (208 B/s)
W: GPG error: http://dl.google.com stable InRelease: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY 78BD65473CB3BD13
W: Failed to fetch http://dl.google.com/linux/chrome/deb/dists/stable/main/binary-i386/Packages 404 Not Found [IP: 2404:6800:4004:813::200e 80]

E: Some index files failed to download. They have been ignored, or old ones used instead.

mkido@mkido-AO531h:~$ apt-cache policy g++-4.8 gcc-4.8
g++-4.8:
  Installed: (none)
  Candidate: 4.8.2-19ubuntu1
  Version table:
     4.8.2-19ubuntu1 0
        500 http://us.archive.ubuntu.com/ubuntu/ trusty/main i386 Packages
gcc-4.8:
  Installed: 4.8.4-2ubuntu1~14.04.4
  Candidate: 4.8.4-2ubuntu1~14.04.4
  Version table:
 *** 4.8.4-2ubuntu1~14.04.4 0
        100 /var/lib/dpkg/status
     4.8.2-19ubuntu1 0
        500 http://us.archive.ubuntu.com/ubuntu/ trusty/main i386 Packages
mkido@mkido-AO531h:~$

END

You are missing trusty-updates in your list of repository sources.
You had it enabled once and you have packages from that repository, but currently it is not enabled any more. This creates the dependency problems.

Open the program software-properties-gtk and re-enable trusty-updates.
This should solve the dependency problem for build-essential.

There is another problem with the google chrome repository.
It seems to me that google provides only the 64bit version and no more the 32bit version.
You have to disable that repository.

Sorry, I missed one:

You should also activate trusty-security.

Hi ! There ! Thank you for wonderful analyses, and proposing answers. I will try to follow your answers in a next few days. One moment please, I will likely feedback here most of my challenges to your proposed answers. Thank you so much. Bye now, till a little while later. From Hachioji, Tokyo.