(t-12) (t+1) =-36

To solve the equation \((t - 12)(t + 1) = -36\), we will first rearrange it into standard quadratic form.

1. Expand the left side of the equation: \[ (t - 12)(t + 1) = t^2 + t - 12t - 12 = t^2 - 11t - 12 \]

2. Set the equation equal to zero by adding 36 to both sides: \[ t^2 - 11t - 12 + 36 = 0 \] \[ t^2 - 11t + 24 = 0 \]

3. Now, we will factor the quadratic equation: \[ t^2 - 11t + 24 = (t - 3)(t - 8) = 0 \]

4. Set each factor to zero: \[ t - 3 = 0 \quad \Rightarrow \quad t = 3 \] \[ t - 8 = 0 \quad \Rightarrow \quad t = 8 \]

Thus, the solutions to the equation \((t - 12)(t + 1) = -36\) are: \[ t = 3 \quad \text{and} \quad t = 8 \]