Factorize Fraction D3A68A

1. **State the problem:** Factorize and simplify the expression $$\frac{5x^2 + 15x}{2x+6} : \left(1 + \frac{5}{2x}\right)$$. 2. **Rewrite the division as multiplication by the reciprocal:** $$\frac{5x^2 + 15x}{2x+6} \times \frac{1}{1 + \frac{5}{2x}}$$ 3. **Factor the numerator and denominator where possible:** - Numerator: $$5x^2 + 15x = 5x(x + 3)$$ - Denominator: $$2x + 6 = 2(x + 3)$$ 4. **Simplify the first fraction:** $$\frac{5x(x + 3)}{2(x + 3)}$$ Cancel the common factor $\cancel{(x + 3)}$: $$\frac{5x\cancel{(x + 3)}}{2\cancel{(x + 3)}} = \frac{5x}{2}$$ 5. **Simplify the second term inside the reciprocal:** $$1 + \frac{5}{2x} = \frac{2x}{2x} + \frac{5}{2x} = \frac{2x + 5}{2x}$$ 6. **Take the reciprocal:** $$\frac{1}{1 + \frac{5}{2x}} = \frac{1}{\frac{2x + 5}{2x}} = \frac{2x}{2x + 5}$$ 7. **Multiply the simplified first fraction by the reciprocal:** $$\frac{5x}{2} \times \frac{2x}{2x + 5} = \frac{5x \times 2x}{2 \times (2x + 5)} = \frac{10x^2}{2(2x + 5)}$$ 8. **Simplify the fraction by canceling common factors:** $$\frac{\cancel{10}x^2}{\cancel{2}(2x + 5)} = \frac{5x^2}{2x + 5}$$ **Final answer:** $$\boxed{\frac{5x^2}{2x + 5}}$$