Simplify the following difference of polynomials completely. \[ \left(u^{2}+v^{2}\right)-\left(-2 u^{2}+5 u v-15 v^{2}\right) \]

To simplify the expression \(\left(u^{2}+v^{2}\right)-\left(-2 u^{2}+5 u v-15 v^{2}\right)\), we start by distributing the negative sign across the second polynomial: \[ u^{2} + v^{2} + 2u^{2} - 5uv + 15v^{2} \] Now, we combine like terms: \[ (u^{2} + 2u^{2}) + (v^{2} + 15v^{2}) - 5uv \] This simplifies to: \[ 3u^{2} + 16v^{2} - 5uv \] Therefore, the completely simplified expression is: \[ 3u^2 - 5uv + 16v^2 \]