1. **State the problem:** Simplify the expression $$\frac{x^2 + 8x + 7 - x}{x} - (3x - 2)$$ and analyze its components. 2. **Rewrite the numerator:** Combine like terms in the numerator: $$x^2 + 8x + 7 - x = x^2 + 7x + 7$$ 3. **Express the fraction:** The expression becomes: $$\frac{x^2 + 7x + 7}{x} - (3x - 2)$$ 4. **Split the fraction:** Divide each term in the numerator by $x$: $$\frac{x^2}{x} + \frac{7x}{x} + \frac{7}{x} - (3x - 2) = x + 7 + \frac{7}{x} - 3x + 2$$ 5. **Combine like terms:** $$x + 7 + \frac{7}{x} - 3x + 2 = (x - 3x) + (7 + 2) + \frac{7}{x} = -2x + 9 + \frac{7}{x}$$ 6. **Final simplified expression:** $$-2x + 9 + \frac{7}{x}$$ This expression is defined for all $x \neq 0$ because of the denominator in the original fraction. **Summary:** The original complex fraction simplifies to $$-2x + 9 + \frac{7}{x}$$ which combines polynomial and rational terms.