1. **State the problem** Factor the expression $9u^2-30uv+25v^2$. 2. **Use the factoring idea (quadratic in $u$)** View it like $au^2+buv+cv^2$ with $a=9$, $b=-30$, $c=25$. 3. **Try a perfect-square/FOIL pattern** Notice $9u^2$ is $\left(3u\right)^2$ and $25v^2$ is $\left(5v\right)^2$. We want $\left(3u-5v\right)^2$. 4. **Check by expanding (FOIL)** $$ \left(3u-5v\right)^2 =\left(3u\right)^2-2\left(3u\right)\left(5v\right)+\left(5v\right)^2 $$ $$ =9u^2-2\cdot 3u\cdot 5v+25v^2 $$ $$ =9u^2-30uv+25v^2 $$ 5. **Final answer** $9u^2-30uv+25v^2=\left(3u-5v\right)^2$.