1. **State the problem:** Simplify the expression $$\frac{8a - 18}{3a^2 + 14a + 8} + \frac{7}{3a + 2}$$. 2. **Factor the denominators and numerator where possible:** - Factor numerator of first fraction: $$8a - 18 = 2(4a - 9)$$. - Factor denominator of first fraction: $$3a^2 + 14a + 8$$. To factor, find two numbers that multiply to $3 \times 8 = 24$ and add to $14$: these are $12$ and $2$. So, $$3a^2 + 14a + 8 = 3a^2 + 12a + 2a + 8 = 3a(a + 4) + 2(a + 4) = (3a + 2)(a + 4)$$. 3. **Rewrite the expression with factored forms:** $$\frac{2(4a - 9)}{(3a + 2)(a + 4)} + \frac{7}{3a + 2}$$. 4. **Find common denominator:** The common denominator is $(3a + 2)(a + 4)$. 5. **Rewrite second fraction with common denominator:** $$\frac{7}{3a + 2} = \frac{7(a + 4)}{(3a + 2)(a + 4)}$$. 6. **Add the fractions:** $$\frac{2(4a - 9)}{(3a + 2)(a + 4)} + \frac{7(a + 4)}{(3a + 2)(a + 4)} = \frac{2(4a - 9) + 7(a + 4)}{(3a + 2)(a + 4)}$$. 7. **Simplify numerator:** $$2(4a - 9) + 7(a + 4) = 8a - 18 + 7a + 28 = (8a + 7a) + (-18 + 28) = 15a + 10$$. 8. **Factor numerator:** $$15a + 10 = 5(3a + 2)$$. 9. **Rewrite the fraction:** $$\frac{5(3a + 2)}{(3a + 2)(a + 4)}$$. 10. **Cancel common factor $(3a + 2)$:** $$\frac{5\cancel{(3a + 2)}}{\cancel{(3a + 2)}(a + 4)} = \frac{5}{a + 4}$$. **Final answer:** $$\boxed{\frac{5}{a + 4}}$$