I have the following expression:
$${2\over x-2} + {2 \over{x^2} -5x +6}$$
So I can simplify this as:
$${2 \over x -2} + {2 \over (x -3) (x-2)}$$
I make the common denominator to be ${(x-3)(x-2)}$
So I then apply ${(x-3)}$ to the left hand side which gives me:
$${2(x-3) + 2 \over (x-3)(x-2)}$$
I have clearly taken a wrong step because the answer in the book to the original expression is ${2 \over x-3}$ so I'm not sure how that answer was arrived at.
On
Notice, the following steps $$\frac{2}{x-2}+\frac{2}{x^2-5x+6}$$ $$=2\left(\frac{1}{x-2}+\frac{1}{(x-2)(x-3)}\right)$$ $$=\frac{2}{x-2}\left(1+\frac{1}{x-3}\right)$$ $$=\frac{2}{x-2}\left(\frac{x-3+1}{x-3}\right)$$ $$=\frac{2}{x-2}\left(\frac{x-2}{x-3}\right)$$ $$=\frac{2}{x-3}$$ Your book has the correct expression.
You did nothing wrong. Note that the numerator can be written as $$2(x-3)+2=2x-6+2=2x-4=2(x-2).$$