Answer
(a) $c = −1$ is a multiple root
(b) $c = −1$ is not a multiple root
Work Step by Step
(a) Consider
$$ f(x)=x^{5}+2 x^{4}-4 x^{3}-8 x^{2}-x+2$$
Since
$$ f'(x)=5x^{4}+8 x^{3}-12x^{2}-16 x-1$$
Then
$$
f(-1)=f'(-1)=0
$$
Hence $c = −1$ is a multiple root of $f(x)$
(b) Given
$$f(x) =x^{4}+x^{3}-5 x^{2}-3 x+2 $$
Since
$$f'(x) =4x^{3}+3x^{2}-10x-3 $$
Then
\begin{align*}
f(-1)&= 0\\
f'(-1)&= 6\neq 0
\end{align*}
Hence $c = −1$ is not multiple root of $f(x)$
