1. **Problem:** Find $f(\frac{3}{2}) + g(-9)$ given $f(x) = 2x - 9$ and $g(x) = 3x^2 - 5x + 2$. 2. **Step 1:** Calculate $f(\frac{3}{2})$. $$f\left(\frac{3}{2}\right) = 2 \times \frac{3}{2} - 9 = 3 - 9 = -6$$ 3. **Step 2:** Calculate $g(-9)$. $$g(-9) = 3(-9)^2 - 5(-9) + 2 = 3 \times 81 + 45 + 2 = 243 + 45 + 2 = 290$$ 4. **Step 3:** Add the results. $$f\left(\frac{3}{2}\right) + g(-9) = -6 + 290 = 284$$ --- 5. **Problem:** Find $\frac{f(-2)}{g(1)} + \frac{1}{2}$. 6. **Step 1:** Calculate $f(-2)$. $$f(-2) = 2(-2) - 9 = -4 - 9 = -13$$ 7. **Step 2:** Calculate $g(1)$. $$g(1) = 3(1)^2 - 5(1) + 2 = 3 - 5 + 2 = 0$$ 8. **Step 3:** Since $g(1) = 0$, division by zero is undefined. **Therefore, $\frac{f(-2)}{g(1)} + \frac{1}{2}$ is undefined.** --- 9. **Problem:** Find $f(2x - 3) - g(x + 4)$. 10. **Step 1:** Calculate $f(2x - 3)$. $$f(2x - 3) = 2(2x - 3) - 9 = 4x - 6 - 9 = 4x - 15$$ 11. **Step 2:** Calculate $g(x + 4)$. $$g(x + 4) = 3(x + 4)^2 - 5(x + 4) + 2$$ Expand $(x + 4)^2$: $$= 3(x^2 + 8x + 16) - 5x - 20 + 2 = 3x^2 + 24x + 48 - 5x - 20 + 2$$ Simplify: $$= 3x^2 + 19x + 30$$ 12. **Step 3:** Subtract $g(x + 4)$ from $f(2x - 3)$. $$f(2x - 3) - g(x + 4) = (4x - 15) - (3x^2 + 19x + 30) = -3x^2 + 4x - 19x - 15 - 30$$ Simplify: $$= -3x^2 - 15x - 45$$ --- **Final answers:** 1. $284$ 2. Undefined (division by zero) 3. $-3x^2 - 15x - 45$ --- **Note:** Only the first problem is fully solved as per instructions.