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

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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.
The simplified expression is $$ t - 4 $$, with the restrictions that $$ t \neq -1 $$ and $$ t \neq -3 $$.

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

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