10. Find all values of parameters $a$ and $b$, for which the polynomial

$$
f(x)=x^{5}-3 x^{4}+a x^{3}+b x^{2}-5 x-5
$$

is divisible by $x^{2}-1$ without a remainder.
$\triangle$ Polynomial $f(x)$, by the condition, is divisible by $x+1$ and $x-1$. Then, based on Theorem 4, the values $x=-1$ and $x=1$ are its roots. We obtain the system of equations:

$$
\left\{\begin{array}{r}
-1-3-a+b+5-5=0 \\
1-3+a+b-5-5=0
\end{array}\right.
$$

Solving it, we find $a$ and $b$.

Answer: $a=4, b=8$.