1. **State the problem:** Simplify the expression $$2 - \frac{3x+2}{x+2} - \frac{2}{7}(2-x)$$. 2. **Rewrite the expression:** $$2 - \frac{3x+2}{x+2} - \frac{2}{7}(2-x)$$ 3. **Distribute the last term:** $$\frac{2}{7}(2-x) = \frac{4}{7} - \frac{2x}{7}$$ So the expression becomes: $$2 - \frac{3x+2}{x+2} - \left(\frac{4}{7} - \frac{2x}{7}\right) = 2 - \frac{3x+2}{x+2} - \frac{4}{7} + \frac{2x}{7}$$ 4. **Combine the constant terms:** $$2 - \frac{4}{7} = \frac{14}{7} - \frac{4}{7} = \frac{10}{7}$$ So now: $$\frac{10}{7} - \frac{3x+2}{x+2} + \frac{2x}{7}$$ 5. **Write all terms with a common denominator to combine:** The denominators are $7$ and $x+2$. The common denominator is $$7(x+2)$$. Rewrite each term: - $$\frac{10}{7} = \frac{10(x+2)}{7(x+2)} = \frac{10x + 20}{7(x+2)}$$ - $$- \frac{3x+2}{x+2} = - \frac{7(3x+2)}{7(x+2)} = - \frac{21x + 14}{7(x+2)}$$ - $$\frac{2x}{7} = \frac{2x(x+2)}{7(x+2)} = \frac{2x^2 + 4x}{7(x+2)}$$ 6. **Combine all numerators:** $$10x + 20 - (21x + 14) + 2x^2 + 4x = 10x + 20 - 21x - 14 + 2x^2 + 4x$$ Simplify: $$2x^2 + (10x - 21x + 4x) + (20 - 14) = 2x^2 - 7x + 6$$ 7. **Final simplified expression:** $$\frac{2x^2 - 7x + 6}{7(x+2)}$$ 8. **Factor numerator if possible:** $$2x^2 - 7x + 6$$ Try factors of $2 \times 6 = 12$ that sum to $-7$: $-3$ and $-4$. Rewrite: $$2x^2 - 3x - 4x + 6 = x(2x - 3) - 2(2x - 3) = (x - 2)(2x - 3)$$ 9. **Final factored form:** $$\frac{(x - 2)(2x - 3)}{7(x + 2)}$$ **Answer:** $$\boxed{\frac{(x - 2)(2x - 3)}{7(x + 2)}}$$