1. **State the problem:** Simplify the expression $$\frac{x^2}{x-3} - \frac{x+15}{2x-6}$$ and express it in the form $$ax + b$$ for all $$x \neq 3$$. Find the value of $$b$$. 2. **Rewrite the denominators:** Note that $$2x - 6 = 2(x-3)$$. 3. **Rewrite the expression with a common denominator:** $$\frac{x^2}{x-3} - \frac{x+15}{2(x-3)} = \frac{2x^2}{2(x-3)} - \frac{x+15}{2(x-3)} = \frac{2x^2 - (x+15)}{2(x-3)}$$ 4. **Simplify the numerator:** $$2x^2 - (x + 15) = 2x^2 - x - 15$$ 5. **Factor the numerator:** We try to factor $$2x^2 - x - 15$$. Find two numbers that multiply to $$2 \times (-15) = -30$$ and add to $$-1$$. These numbers are $$5$$ and $$-6$$. Rewrite: $$2x^2 - x - 15 = 2x^2 + 5x - 6x - 15 = (2x^2 + 5x) - (6x + 15) = x(2x + 5) - 3(2x + 5) = (x - 3)(2x + 5)$$ 6. **Substitute back:** $$\frac{(x - 3)(2x + 5)}{2(x - 3)}$$ 7. **Cancel common factors:** Since $$x \neq 3$$, $$x - 3 \neq 0$$, so $$\frac{(x - 3)(2x + 5)}{2(x - 3)} = \frac{2x + 5}{2} = x + \frac{5}{2}$$ 8. **Identify constants:** The expression is equivalent to $$ax + b$$ where $$a = 1$$ and $$b = \frac{5}{2}$$. **Final answer:** $$b = \frac{5}{2}$$. **Answer choice:** b. 5/2