21. \( \frac{t^{2}-t-12}{t+1} \cdot \frac{t+1}{t+3} \)

To simplify the expression: \[ \frac{t^{2} - t - 12}{t + 1} \cdot \frac{t + 1}{t + 3} \] **Step 1: Factor the Quadratic** First, factor the quadratic expression in the numerator: \[ t^{2} - t - 12 \] We look for two numbers that multiply to \(-12\) and add to \(-1\). These numbers are \(-4\) and \(3\). \[ t^{2} - t - 12 = (t - 4)(t + 3) \] **Step 2: Rewrite the Expression with Factored Form** Substitute the factored form back into the original expression: \[ \frac{(t - 4)(t + 3)}{t + 1} \cdot \frac{t + 1}{t + 3} \] **Step 3: Simplify by Cancelling Common Factors** Notice that \((t + 1)\) and \((t + 3)\) appear in both the numerator and the denominator. Assuming \(t \neq -1\) and \(t \neq -3\) to avoid division by zero, we can cancel these terms: \[ \frac{\cancel{(t - 4)}\cancel{(t + 3)}}{\cancel{t + 1}} \cdot \frac{\cancel{t + 1}}{\cancel{t + 3}} = t - 4 \] **Final Simplified Expression:** \[ t - 4 \] **Restrictions:** - \( t \neq -1 \) - \( t \neq -3 \) These restrictions ensure that the original expression is defined.