Question 173
Suppose $a$ and $b$ are real numbers such that the quadratic polynomial
$$
f(x) = x^2 + a x + b
$$
has no nonnegative real roots. Prove that there exist two polynomials $g, h$, whose coefficients are nonnegative real numbers, such that
$$
f(x) = \frac{g(x)}{h(x)},
$$
for all real numbers $x$.
$$
f(x) = x^2 + a x + b
$$
has no nonnegative real roots. Prove that there exist two polynomials $g, h$, whose coefficients are nonnegative real numbers, such that
$$
f(x) = \frac{g(x)}{h(x)},
$$
for all real numbers $x$.