To factor the expression \( u(u - v) + v(v - u) \), let's start by rewriting it:
\[ u(u - v) + v(v - u) = u^2 - uv + v^2 - vu \]
Notice that \( -uv \) and \( -vu \) are the same term. Therefore, we can simplify further:
\[ u^2 - uv + v^2 - uv = u^2 - 2uv + v^2 \]
Now we recognize that \( u^2 - 2uv + v^2 \) can be factored as a perfect square:
\[ u^2 - 2uv + v^2 = (u - v)^2 \]
So, the original expression \( u(u - v) + v(v - u) \) factors to:
\[ (u - v)^2 \]
Thus, the final factored form is:
\[ \boxed{(u - v)^2} \]