Simplify 15x2 - 15y2 / 5x2 + 10xy + 5y2

The numerator is $15x^2 - 15y^2$. We can factor out the common term 15.

$$15x^2 - 15y^2 = 15(x^2 - y^2)$$

Recognize that $x^2 - y^2$ is a difference of squares, which factors as $(x-y)(x+y)$.
So, the numerator becomes:

$$15(x-y)(x+y)$$

The denominator is $5x^2 + 10xy + 5y^2$. We can factor out the common term 5.

$$5x^2 + 10xy + 5y^2 = 5(x^2 + 2xy + y^2)$$

Recognize that $x^2 + 2xy + y^2$ is a perfect square trinomial, which factors as $(x+y)^2$.
So, the denominator becomes:

$$5(x+y)^2$$

Divide 15 by 5: $\frac{15}{5} = 3$.
Cancel one $(x+y)$ term from the numerator and the denominator: $\frac{x+y}{(x+y)^2} = \frac{1}{x+y}$.

$$\frac{3(x-y)}{x+y}$$