1. **State the problem:** Simplify the expression $$\frac{x^{2} - 16}{x^{2} - 8x + 16} - \frac{x^{2}}{x^{2} + 4x} = \lambda$$ and express it in simplest form. 2. **Factor all polynomials:** - Numerator of first fraction: $$x^{2} - 16 = (x - 4)(x + 4)$$ - Denominator of first fraction: $$x^{2} - 8x + 16 = (x - 4)^{2}$$ - Numerator of second fraction: $$x^{2}$$ (already factored as $$x \cdot x$$) - Denominator of second fraction: $$x^{2} + 4x = x(x + 4)$$ 3. **Rewrite the expression with factored forms:** $$\frac{(x - 4)(x + 4)}{(x - 4)^{2}} - \frac{x^{2}}{x(x + 4)}$$ 4. **Simplify each fraction where possible:** - First fraction: cancel one $$x - 4$$ term: $$\frac{(x - 4)(x + 4)}{(x - 4)^{2}} = \frac{x + 4}{x - 4}$$ - Second fraction: simplify numerator and denominator: $$\frac{x^{2}}{x(x + 4)} = \frac{x}{x + 4}$$ 5. **Rewrite the expression:** $$\frac{x + 4}{x - 4} - \frac{x}{x + 4}$$ 6. **Find common denominator:** The common denominator is $$(x - 4)(x + 4)$$. 7. **Rewrite each fraction with common denominator:** $$\frac{(x + 4)^{2}}{(x - 4)(x + 4)} - \frac{x(x - 4)}{(x + 4)(x - 4)}$$ 8. **Combine the fractions:** $$\frac{(x + 4)^{2} - x(x - 4)}{(x - 4)(x + 4)}$$ 9. **Expand the numerators:** - $$(x + 4)^{2} = x^{2} + 8x + 16$$ - $$x(x - 4) = x^{2} - 4x$$ 10. **Subtract the numerators:** $$x^{2} + 8x + 16 - (x^{2} - 4x) = x^{2} + 8x + 16 - x^{2} + 4x = 12x + 16$$ 11. **Simplify numerator:** $$12x + 16 = 4(3x + 4)$$ 12. **Rewrite the expression:** $$\frac{4(3x + 4)}{(x - 4)(x + 4)}$$ 13. **Final simplified form:** $$\lambda = \frac{4(3x + 4)}{x^{2} - 16}$$ **Answer:** $$\boxed{\lambda = \frac{4(3x + 4)}{x^{2} - 16}}$$