Who can prove that $7|4x^2+5x+2+d $ following solutions I found are correct (complete)? Where $d$ is integer constant (not relation with $x$), find $d$ which make $7|4x^2+5x+2+d $ have solution, and give corresponding solution $x$ as well.
I have found following would be correct solutions, but not sure if them are complete: $d≡0~ mod~7~⟹~ (x=2+7t)$; $d≡3~ mod~7~⟹~ (x=1+7t, x=3+7t)$; $d≡5~ mod~7~⟹~ (x=4+7t, x=7t)$; $d≡6~ mod~7~⟹~ (x=5+7t, x=6+7t)$.
My approach is find them by testing $d$ value one by one by, then I can get that $d$ with pattern $d≡{0,3, 5, 6}~ mod ~7$ and the copressponding $x$ can be sloved, but I couldn't prove that these are complete solutions.
A more genral question is when $ pq|ax^2+bx+c$ has solution, where $pq$ is semiprime, $a,b,c$ are integer constant, find $x$. If not easy to answer, do we have a general algorithm within limit steps to detect any given $pq, a,b,c$ for $ pq|ax^2+bx+c $ has solution or has no solution?