ify the following difference of polynomials completely. \[ \left(s^{2}+t^{2}\right)-\left(-15 s^{2}+2 s t-8 t^{2}\right) \]

We start with the expression \[ \left(s^{2}+t^{2}\right)-\left(-15 s^{2}+2 s t-8 t^{2}\right). \] **Step 1. Distribute the negative sign:** Subtracting the second polynomial means we change the sign of each of its terms: \[ s^{2}+t^{2} +15 s^{2}-2 s t+8 t^{2}. \] **Step 2. Combine like terms:** Combine the \(s^2\) terms: \[ s^2+15 s^2 = 16 s^2. \] The \(st\) term remains: \[ -2 s t. \] Combine the \(t^2\) terms: \[ t^2+8t^2 = 9 t^2. \] So the simplified expression is: \[ 16 s^2-2 s t+9 t^2. \] **Step 3. Check for further factoring:** The expression \(16 s^2-2 s t+9 t^2\) is a quadratic in \(s\) (or \(t\)). Its discriminant in terms of \(s\) is given by \[ \Delta = (-2t)^2-4\cdot16\cdot9t^2=4t^2-576t^2=-572t^2. \] Since \(\Delta<0\) (for any nonzero \(t\)), the quadratic does not factor further over the real numbers. Thus, the fully factored form of the original expression is: \[ 16 s^2-2 s t+9 t^2. \]