[ Factorization $]$

Can the polynomial $x^{4}+x^{3}+x^{2}+x+12$ be factored with integer coefficients?
Let's try to represent the given polynomial as $\left(x^{2}+a x+3\right)\left(x^{2}+b x+4\right)$. Expanding the brackets, we get the relations

$a+b=1, a b+3+4=1, 4 a+3 b=1$. Subtracting three times the first equation from the third, we get $a=-2$.

From this, $b=3$, which satisfies the second relation as well. Therefore, $x^{4}+x^{3}+x^{2}+x+12=\left(x^{2}-2 x+3\right)\left(x^{2}+3 x+4\right)$.

## Answer

It can be done.