1. **State the problem:** Simplify the expression $$\frac{7x}{12} + \frac{x-2}{6} - \frac{x^2}{3}$$. 2. **Find a common denominator:** The denominators are 12, 6, and 3. The least common denominator (LCD) is 12. 3. **Rewrite each fraction with denominator 12:** - $$\frac{7x}{12}$$ stays the same. - $$\frac{x-2}{6} = \frac{(x-2) \times 2}{6 \times 2} = \frac{2(x-2)}{12}$$. - $$\frac{x^2}{3} = \frac{x^2 \times 4}{3 \times 4} = \frac{4x^2}{12}$$. 4. **Rewrite the expression:** $$\frac{7x}{12} + \frac{2(x-2)}{12} - \frac{4x^2}{12}$$ 5. **Combine the fractions over the common denominator:** $$\frac{7x + 2(x-2) - 4x^2}{12}$$ 6. **Expand the numerator:** $$7x + 2x - 4 - 4x^2 = (7x + 2x) - 4 - 4x^2 = 9x - 4 - 4x^2$$ 7. **Rewrite numerator in standard polynomial form:** $$-4x^2 + 9x - 4$$ 8. **Final simplified expression:** $$\frac{-4x^2 + 9x - 4}{12}$$ 9. **Check for common factors:** The numerator coefficients are -4, 9, and -4. No common factor with 12 except 1, so expression is simplified. **Answer:** $$\boxed{\frac{-4x^2 + 9x - 4}{12}}$$