22. Simplify the expression: $$\frac{9x^2 + 90xy + 225y^2}{225y^2 - 9x^2}$$

Solution:

1. Factor the numerator: The numerator is a perfect square trinomial. First, factor out the greatest common divisor, which is 9: $$9(x^2 + 10xy + 25y^2)$$. The expression inside the parentheses is a perfect square trinomial: $$x^2 + 10xy + 25y^2 = (x + 5y)^2$$. So the numerator is $$9(x + 5y)^2$$.
2. Factor the denominator: The denominator is a difference of squares: $$225y^2 - 9x^2 = (15y)^2 - (3x)^2 = (15y - 3x)(15y + 3x)$$.
3. Factor out common terms in the denominator: From $$(15y - 3x)$$, we can factor out 3: $$3(5y - x)$$. From $$(15y + 3x)$$, we can factor out 3: $$3(5y + x)$$. So the denominator is $$3(5y - x) \times 3(5y + x) = 9(5y - x)(5y + x)$$.
4. Simplify the fraction: $$\frac{9(x + 5y)^2}{9(5y - x)(5y + x)}$$. The 9s cancel out.
5. Rewrite the expression: $$\frac{(x + 5y)^2}{(5y - x)(5y + x)}$$.

Answer: $$\frac{(x + 5y)^2}{(5y - x)(5y + x)}$$