Answer
$2$ is an upper bound for the real zeros of $P$
$-4$ is a lower bound for the real zeros of $P$
Work Step by Step
Suppose $f$ is a polynomial of degree $n\geq 1$.
If $c>0$ is synthetically divided into $f$ and all of the coefficients of the quotient and remainder are all non-negative values, then $c$ is an upper bound for the real zeros of $f$. That is, there are no real zeros greater than $c$.
If $c<0$ is synthetically divided into $f$ and the coefficients of the quotient and remainder alternate signs, then $c$ is a lower bound for the real zeros of $f$. That is, there are no real zeros less than $c$.
$P(x)=x^4+3x^3-4x^2-2x-7$; $a=-4$, $b=2$
$\begin{array}{lllll}
\underline{2}| & 1&3 & -4 & -2 & -7& \\
& &2& 10 & 12 & 20\\
\hline & & & & \\
& 1&5 & 6 & 10 & 13
\end{array}$
All of the coefficients of the quotient and remainder are all non-negative values. thus, $2$ is an upper bound for the real zeros of $P$.
$\begin{array}{lllll}
\underline{-4}| &1& 3 & -4 & -2 & -7\\
& & -4&4 & 0& 6\\
\hline & & & & \\
& 1&-1& 0 & -2 & -1
\end{array}$
The coefficients of the quotient and remainder alternates signs. thus, $-4$ is a lower bound for the real zeros of $P$.
