Is the following statement is true/false?
Consider the polynomial $$f(x)=x^4-x^3+14x^2+5x+16,$$ then $f$ is a product of two polynomials of degree two over $\mathbb{Z}$.
My answer : I think it will be true $f(x)=x^4-x^3+14x^2+5x+16= (x^2 +ax+b)(x^2 +cx + d)$
Is its True ?
Any hints/solution will be appreciated
Hint: the image of $f(x)$ in $(\mathbb{Z} / 2 \mathbb{Z})[x]$ is $x^4 + x^3 + x = x(x^3 + x^2 + 1)$. However, $x^3 + x^2 + 1$ is irreducible in this ring (why?). Can you complete the proof to show that this implies that the original polynomial cannot be a product of two quadratic polynomials in $\mathbb{Z}[x]$?