Question 16
Let $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$ be nonzero real numbers. Prove that the polynomial
$$
P(X) = \prod_{k=0}^{4} \left( a_{k+1} X^{4} + a_{k+2} X^{3} + a_{k+3} X^{2} + a_{k+4} X + a_{k+5} \right),
$$
where $a_{5+i} = a_{i}$ for $i = 1, 2, 3, 4$, has a root with negative real part.
$$
P(X) = \prod_{k=0}^{4} \left( a_{k+1} X^{4} + a_{k+2} X^{3} + a_{k+3} X^{2} + a_{k+4} X + a_{k+5} \right),
$$
where $a_{5+i} = a_{i}$ for $i = 1, 2, 3, 4$, has a root with negative real part.