Answer
1. Upper bound $b=5$, see prove below.
2. Lower bound $a=-3$, see prove below.
Work Step by Step
1. Upper bound $b=5$, we use synthetic division to divide the polynomial by $x-5$ as shown in the figure.
As can be seen, all the numbers $1,3,6,32,168$ in the line containing the quotient and remainder of this division are positive. Based on the Upper and Lower Bound Theorem (page 279 in the book), we conclude that $b=5$ is an upper bound of the function $P(x)$.
2. Lower bound $a=-3$, similarly, we used synthetic division to divide $P(x)$ by $x+3$ as shown in the figure,
and it can be seen that the numbers $1,-5,6,-16,56$ on the quotient and remainder line have alternating signs.
Based on the Upper and Lower Bound Theorem (page 279 in the book), we conclude that $a=-3$ is a lower bound of the function $P(x)$.
