KöMaL 2016 B4821

Does there exist an integer $a\ge 1$ such that $x^2+3$ and ${(x+a)}^2+3$ are relatively primes for all positive integers $x$ ?

Official text KöMaL. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.