We are given the function $f(x) = (x-a)^2(x-b)^2$ and we need to find the value of $f(a+b)$.

1. Problem Description

We are given the function $f(x) = (x-a)^2(x-b)^2$ and we need to find the value of $f(a+b)$.

2. Solution Steps

To find $f(a+b)$, we need to substitute $x$ with $a+b$ in the given function.

$f(a+b) = ((a+b)-a)^2((a+b)-b)^2$

$f(a+b) = (a+b-a)^2(a+b-b)^2$

$f(a+b) = (b)^2(a)^2$

$f(a+b) = a^2b^2$

3. Final Answer

$f(a+b) = a^2b^2$