1. **State the problem:** Simplify the expression $$\frac{6x}{x^2 - 5x + 6} - \frac{3x}{x^2 + x - 12}$$ and state any restrictions on the variables. 2. **Factor the denominators:** - Factor $$x^2 - 5x + 6$$ as $$(x - 2)(x - 3)$$ because $-2 \times -3 = 6$ and $-2 + -3 = -5$. - Factor $$x^2 + x - 12$$ as $$(x + 4)(x - 3)$$ because $4 \times -3 = -12$ and $4 + (-3) = 1$. 3. **Rewrite the expression with factored denominators:** $$\frac{6x}{(x - 2)(x - 3)} - \frac{3x}{(x + 4)(x - 3)}$$ 4. **Find the common denominator:** The common denominator is $$(x - 2)(x - 3)(x + 4)$$. 5. **Rewrite each fraction with the common denominator:** $$\frac{6x(x + 4)}{(x - 2)(x - 3)(x + 4)} - \frac{3x(x - 2)}{(x + 4)(x - 3)(x - 2)}$$ 6. **Combine the fractions:** $$\frac{6x(x + 4) - 3x(x - 2)}{(x - 2)(x - 3)(x + 4)}$$ 7. **Expand the numerators:** $$6x(x + 4) = 6x^2 + 24x$$ $$3x(x - 2) = 3x^2 - 6x$$ 8. **Substitute back and simplify numerator:** $$6x^2 + 24x - (3x^2 - 6x) = 6x^2 + 24x - 3x^2 + 6x = (6x^2 - 3x^2) + (24x + 6x) = 3x^2 + 30x$$ 9. **Factor numerator:** $$3x^2 + 30x = 3x(x + 10)$$ 10. **Final simplified expression:** $$\frac{3x(x + 10)}{(x - 2)(x - 3)(x + 4)}$$ 11. **State restrictions:** The denominators cannot be zero, so: - $$x - 2 \neq 0 \Rightarrow x \neq 2$$ - $$x - 3 \neq 0 \Rightarrow x \neq 3$$ - $$x + 4 \neq 0 \Rightarrow x \neq -4$$ **Answer:** $$\boxed{\frac{3x(x + 10)}{(x - 2)(x - 3)(x + 4)}, \quad x \neq 2, 3, -4}$$