Answer
1. Upper bound $b=6$, see proof below.
2. Lower bound $a=0$, see proof below.
Work Step by Step
1. Upper bound $b=6$, we use synthetic division to divide the polynomial by $x-6$ as shown in the figure. As can be seen, all the numbers $3,1,30,171,1027$ 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 $b=6$ is an upper bound of the function $P(x)$.
2. Lower bound $a=0$, similarly, we used synthetic division to divide $P(x)$ by $x$ as shown in the figure, and it can be seen that the numbers $3,-17,24,-9,1$ 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 $a=0$ is a lower bound of the function $P(x)$.
