Question: 4) $\frac{3x^2 - x - 2}{2x - 2}$
5) $\frac{(a^2 - 4)(a + 2)}{a - 2}$
1. **Problem 4:** Simplify the expression $$\frac{3x^2 - x - 2}{2x - 2}$$.
2. **Step 1:** Factor numerator and denominator.
- Numerator: $3x^2 - x - 2$
- Find two numbers that multiply to $3 \times (-2) = -6$ and add to $-1$.
- These numbers are $-3$ and $2$.
- Rewrite: $3x^2 - 3x + 2x - 2$
- Factor by grouping: $3x(x - 1) + 2(x - 1) = (3x + 2)(x - 1)$.
- Denominator: $2x - 2 = 2(x - 1)$.
3. **Step 2:** Write the fraction with factored terms:
$$\frac{(3x + 2)(x - 1)}{2(x - 1)}$$
4. **Step 3:** Cancel the common factor $(x - 1)$:
$$\frac{(3x + 2)\cancel{(x - 1)}}{2\cancel{(x - 1)}} = \frac{3x + 2}{2}$$
5. **Answer for Problem 4:** $$\frac{3x + 2}{2}$$
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6. **Problem 5:** Simplify the expression $$\frac{(a^2 - 4)(a + 2)}{a - 2}$$.
7. **Step 1:** Factor $a^2 - 4$ as a difference of squares:
$$a^2 - 4 = (a - 2)(a + 2)$$
8. **Step 2:** Substitute back:
$$\frac{(a - 2)(a + 2)(a + 2)}{a - 2} = \frac{(a - 2)(a + 2)^2}{a - 2}$$
9. **Step 3:** Cancel the common factor $(a - 2)$:
$$\frac{\cancel{(a - 2)}(a + 2)^2}{\cancel{(a - 2)}} = (a + 2)^2$$
10. **Answer for Problem 5:** $$(a + 2)^2$$