Simplify the following difference of polynomials eompletely, \[ \left(A^{2}+t^{2}\right)-\left(-15 B^{2}+2 A t-8 t^{2}\right) \]

To simplify the given expression, we start by distributing the negative sign across the second polynomial: \[ \left(A^{2} + t^{2}\right) - \left(-15 B^{2} + 2 A t - 8 t^{2}\right) = A^{2} + t^{2} + 15 B^{2} - 2 A t + 8 t^{2} \] Next, we combine like terms: 1. The \( t^{2} \) terms: \( t^{2} + 8 t^{2} = 9 t^{2} \) 2. The other terms stay as they are: \( A^{2}, 15 B^{2}, -2 A t \) Now we can write the simplified expression: \[ A^{2} + 15 B^{2} - 2 A t + 9 t^{2} \] So, the final simplified expression is: \[ A^{2} + 15 B^{2} - 2 A t + 9 t^{2} \]