1. **State the problem:** Simplify the expression $$2 - \frac{x + 2}{x - 3} - \frac{x - 6}{x + 3}$$ into a single fraction of the form $$\frac{ax + b}{x^2 - 9}$$ where $a$ and $b$ are integers. 2. **Identify the common denominator:** The denominators are $x - 3$ and $x + 3$. Their product is $$ (x - 3)(x + 3) = x^2 - 9 $$. 3. **Rewrite each term with the common denominator:** - The term $2$ can be written as $$\frac{2(x^2 - 9)}{x^2 - 9}$$. - The term $$- \frac{x + 2}{x - 3}$$ can be rewritten as $$- \frac{(x + 2)(x + 3)}{x^2 - 9}$$. - The term $$- \frac{x - 6}{x + 3}$$ can be rewritten as $$- \frac{(x - 6)(x - 3)}{x^2 - 9}$$. 4. **Combine all terms over the common denominator:** $$\frac{2(x^2 - 9) - (x + 2)(x + 3) - (x - 6)(x - 3)}{x^2 - 9}$$ 5. **Expand the numerators:** - $$2(x^2 - 9) = 2x^2 - 18$$ - $$(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$$ - $$(x - 6)(x - 3) = x^2 - 3x - 6x + 18 = x^2 - 9x + 18$$ 6. **Substitute back and simplify numerator:** $$2x^2 - 18 - (x^2 + 5x + 6) - (x^2 - 9x + 18)$$ $$= 2x^2 - 18 - x^2 - 5x - 6 - x^2 + 9x - 18$$ $$= (2x^2 - x^2 - x^2) + (-5x + 9x) + (-18 - 6 - 18)$$ $$= 0x^2 + 4x - 42$$ 7. **Final simplified form:** $$\frac{4x - 42}{x^2 - 9}$$ Thus, $a = 4$ and $b = -42$. **Answer:** $$\boxed{\frac{4x - 42}{x^2 - 9}}$$