Question

How did my solutions manual simplify
3(x^2 - y^2)/x^2 + 2x + yx + 2y to
3(x + y)(x - y)/ (x+2)(x+y)? I can't figure out how they simplified the denominator to what they did..

Answer 1

notice that x^2 +2x = x(x+2)
and yx+2y = y(x+2)

Answer 2

\[\frac{3(x^2-y^2)}{x^2+2x+yx+2y}=\frac{3(x+y)(x-y)}{[x(x+2)+y(x+2)]}\] Now the x+2 is common on the denominator so you get \[\frac{3(x+y)(x-y)}{(x+y)(x+2)}\] For the numerator, notice that \[(x+y)(x-y)=x(x-y) +y(x-y)=x^2-xy+xy-y^2=x^2-y^2\]