1. **State the problem:** Simplify the expression $$A = \frac{3x^2 - 12x + 12}{x^2 - 5x + 6} \cdot \frac{6x^3 - 54x}{x^3 - 6x^2 + 9x} - \frac{x^2 + x - 6}{2(x^2 - 9)}$$ 2. **Factor all polynomials:** - Numerator and denominator of first fraction: $$3x^2 - 12x + 12 = 3(x^2 - 4x + 4) = 3(x-2)^2$$ $$x^2 - 5x + 6 = (x-2)(x-3)$$ - Numerator and denominator of second fraction: $$6x^3 - 54x = 6x(x^2 - 9) = 6x(x-3)(x+3)$$ $$x^3 - 6x^2 + 9x = x(x^2 - 6x + 9) = x(x-3)^2$$ - Numerator and denominator of third fraction: $$x^2 + x - 6 = (x+3)(x-2)$$ $$x^2 - 9 = (x-3)(x+3)$$ 3. **Rewrite the expression with factored forms:** $$A = \frac{3(x-2)^2}{(x-2)(x-3)} \cdot \frac{6x(x-3)(x+3)}{x(x-3)^2} - \frac{(x+3)(x-2)}{2(x-3)(x+3)}$$ 4. **Simplify each fraction by canceling common factors:** - First fraction: $$\frac{3\cancel{(x-2)}(x-2)}{\cancel{(x-2)}(x-3)} = \frac{3(x-2)}{x-3}$$ - Second fraction: $$\frac{6x(x-3)(x+3)}{x\cancel{(x-3)^2}} = \frac{6x(x-3)(x+3)}{x(x-3)(x-3)}$$ Cancel $x$ and one $(x-3)$: $$\frac{6\cancel{x}(x-3)(x+3)}{\cancel{x}(x-3)\cancel{(x-3)}} = \frac{6(x+3)}{x-3}$$ 5. **Multiply the simplified first and second fractions:** $$\frac{3(x-2)}{x-3} \cdot \frac{6(x+3)}{x-3} = \frac{18(x-2)(x+3)}{(x-3)^2}$$ 6. **Simplify the third fraction:** $$\frac{(x+3)(x-2)}{2(x-3)(x+3)} = \frac{\cancel{(x+3)}(x-2)}{2(x-3)\cancel{(x+3)}} = \frac{x-2}{2(x-3)}$$ 7. **Rewrite the entire expression:** $$A = \frac{18(x-2)(x+3)}{(x-3)^2} - \frac{x-2}{2(x-3)}$$ 8. **Find common denominator $(x-3)^2$ for subtraction:** Rewrite second term: $$\frac{x-2}{2(x-3)} = \frac{(x-2)(x-3)}{2(x-3)^2}$$ 9. **Express both terms with denominator $(x-3)^2$:** $$A = \frac{18(x-2)(x+3)}{(x-3)^2} - \frac{(x-2)(x-3)}{2(x-3)^2}$$ 10. **Combine the fractions:** $$A = \frac{36(x-2)(x+3)}{2(x-3)^2} - \frac{(x-2)(x-3)}{2(x-3)^2} = \frac{36(x-2)(x+3) - (x-2)(x-3)}{2(x-3)^2}$$ 11. **Factor out $(x-2)$ in numerator:** $$= \frac{(x-2)[36(x+3) - (x-3)]}{2(x-3)^2}$$ 12. **Simplify inside the bracket:** $$36(x+3) - (x-3) = 36x + 108 - x + 3 = 35x + 111$$ 13. **Final simplified expression:** $$A = \frac{(x-2)(35x + 111)}{2(x-3)^2}$$ **Answer:** $$\boxed{\frac{(x-2)(35x + 111)}{2(x-3)^2}}$$