Simplification of rational expressions

I have the following expression:

$\frac{2}{x-2}+\frac{2}{{x}^{2}-5x+6}$

So I can simplify this as:

$\frac{2}{x-2}+\frac{2}{(x-3)(x-2)}$

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:

$\frac{2(x-3)+2}{(x-3)(x-2)}$

I have clearly taken a wrong step because the answer in the book to the original expression is $\frac{2}{x-3}$ so I'm not sure how that answer was arrived at.

I have the following expression:

$\frac{2}{x-2}+\frac{2}{{x}^{2}-5x+6}$

So I can simplify this as:

$\frac{2}{x-2}+\frac{2}{(x-3)(x-2)}$

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:

$\frac{2(x-3)+2}{(x-3)(x-2)}$

I have clearly taken a wrong step because the answer in the book to the original expression is $\frac{2}{x-3}$ so I'm not sure how that answer was arrived at.