\( 2 r(t+3)+t^{2}=(t-5)^{2} \quad \) for \( t \)

To solve the equation \( 2 r(t+3) + t^{2} = (t-5)^{2} \), we first expand the right side: \[ (t-5)^{2} = t^{2} - 10t + 25. \] Now, substituting this back into the equation gives us: \[ 2 r(t+3) + t^{2} = t^{2} - 10t + 25. \] Next, we can eliminate \( t^{2} \) from both sides: \[ 2 r(t+3) = -10t + 25. \] Now, let's isolate \( r(t+3) \): \[ r(t+3) = \frac{-10t + 25}{2}. \] To further simplify, assume \( r \) could be a constant or a specific function we define later. This sets the foundation for solving for \( t \) depending on the value of \( r \). If \( r \) is constant, then we can represent \( r \) in terms of \( t \) or vice versa to find valid solutions that satisfy the equation. If you have a specific value or definition for \( r \), you can replace it here and proceed with solving for \( t \) directly!