### Instruction: Find the remainder when $x^5-x^4-x^3+x^2+x$ is divided by $(x^2-4)(x+1)$.### Response: I know that I can use the remainder theorem to find the remainder when a polynomial $f(x)$ is divided by another polynomial $g(x)$.
The remainder theorem says that the remainder is equal to $f(a)$, where $a$ is any value that makes $g(x)=0$.
In this case, $g(x) = (x^2-4)(x+1)$, so I can choose any of the values $x=2,-2,-1$ to plug into $f(x)$ and get the same remainder.
I think I will choose $x=-1$, because it seems like the simplest option.
So I need to find $f(-1)$, which is the same as plugging in $x=-1$ into the original polynomial.
I get $f(-1) = (-1)^5-(-1)^4-(-1)^3+(-1)^2+(-1)$.
Simplifying the powers of $-1$, I get $f(-1) = -1-1+1+1-1$.
Combining the terms, I get $f(-1) = -1$.
Therefore, the remainder when $x^5-x^4-x^3+x^2+x$ is divided by $(x^2-4)(x+1)$ is $-1$.