1. **State the problem:** Simplify the expression $$\frac{x^4 + 7x^3}{2x^2 + 14x}$$. 2. **Identify common factors:** - Numerator: $$x^4 + 7x^3 = x^3(x + 7)$$ - Denominator: $$2x^2 + 14x = 2x(x + 7)$$ 3. **Rewrite the fraction using factored forms:** $$\frac{x^3(x + 7)}{2x(x + 7)}$$ 4. **Cancel common factors:** Since $$x + 7$$ appears in both numerator and denominator, and $$x$$ appears in both, we can cancel them: $$\frac{\cancel{x^3}(\cancel{x + 7})}{2\cancel{x}(\cancel{x + 7})} = \frac{x^{3-1}}{2} = \frac{x^2}{2}$$ 5. **Final simplified expression:** $$\frac{x^2}{2}$$ This means the original fraction simplifies to $$\frac{x^2}{2}$$ for all $$x \neq 0$$ and $$x \neq -7$$ (to avoid division by zero).