So first of all if there exist such 3 non-constant polynomials then $P(x)$ = $P_1(x)$$P_2(x)$$P_3(x)$ but every $P(x) > 0$ has no real root. Therefore $P(x)$ is divisible by only polynomials with even degree. Since deg $P(x)$ = $8$ $\Rightarrow$ deg $P_1(x)$ = deg $P_2(x)$ = $2$ and deg $P_3(x)$ = $4$. ($P_1(x)$, $P_2(x)$ and $P_3(x)$ has no real roots) $P_1(x)$ = $x^2$ + $ax$ +$b$ and $P_2(x)$ = $x^2$ + $cx$ + $d$ We have $P(x)$ = $(x^2-12x+11)^4$ + $23$ $\Rightarrow$ $P(1)$ = $P_1(1)$$P_2(1)$$P_3(1)$ = $23$ and since $23$ is a prime $\Rightarrow$ $P_1(1)$ or $P_2(1)$ = 1 $P_1(1)$ = 1 $\Rightarrow$ $a$+$b$ = $0$ $P_1(11)$ = $121$ + $11a$ + $b$ = $121$ + $10a$ If $121$ + $10a$ = $23$ $\Rightarrow$ $a$ = $-98/10$ (which is wrong since the polynomial has integer coefficients) $\Rightarrow$ $121$ + $10a$ = $1$ $\Rightarrow$ $a$ = $-12$ and $b$ = $12$ then we have $P_1(x)$ = $x^2$-$12x$+$12$ which has real roots (which contradicts ) So $P(x)$ = $(x^2-12x+11)^4$ + $23$ can not be writte as the product of three non-constant polynomials with integer coefficients. This problem is kinda similar to P2 VMO 2014 which is Prove that the polynomial $P(x)$ = $(x^2-7x+6)^{2n}$ + $13$ can not be written as the porduct of $n$ + $1$ non-constant polynomials with integer coefficients.