Existence of polynomials $g$, $h$, with non-negative coefficients, such that $f(x)= \frac{g(x)}{h(x)}$ Suppose $a$ and $b$ are real numbers such that the quadratic polynomial $f(x) = x^2 + ax + b$has no non-negative real roots. Prove that ther exist two polynomials g,h, whose coefficients are non-negative real numbers, such that
$$f(x)= \frac{g(x)}{h(x)}$$
for all real numbers $x$.
Irish MO 2007

Let
$$
g(x)=f(x)
$$
$$
h(x)=1
$$
