1. **State the problem:** Simplify the expression $$\frac{-4}{x^2 + 2x} + \frac{x}{x + 2}$$. 2. **Rewrite the denominator:** Note that $$x^2 + 2x = x(x + 2)$$. 3. **Rewrite the expression:** $$\frac{-4}{x(x + 2)} + \frac{x}{x + 2}$$ 4. **Find a common denominator:** The common denominator is $$x(x + 2)$$. 5. **Rewrite each term with the common denominator:** $$\frac{-4}{x(x + 2)} + \frac{x \cdot x}{x(x + 2)} = \frac{-4}{x(x + 2)} + \frac{x^2}{x(x + 2)}$$ 6. **Combine the fractions:** $$\frac{-4 + x^2}{x(x + 2)}$$ 7. **Simplify the numerator:** $$x^2 - 4$$ is a difference of squares, so $$x^2 - 4 = (x - 2)(x + 2)$$ 8. **Rewrite the expression:** $$\frac{(x - 2)(x + 2)}{x(x + 2)}$$ 9. **Cancel common factors:** $$\frac{(x - 2)\cancel{(x + 2)}}{x\cancel{(x + 2)}} = \frac{x - 2}{x}$$ **Final answer:** $$\boxed{\frac{x - 2}{x}}$$