Hey, DeathlyHallows!
It looks like you want to simplify \(\frac{x^2-5x+6}{\textcolor{red}{2x-1+\frac{x}{2}}}\) . Let's transform all the terms in the denominator such that they all have common denominators; this way, we will be able to add them together.
\(\textcolor{red}{2x-1+\frac{x}{2}}\Rightarrow\textcolor{red}{\frac{4x}{2}-\frac{1}{2}+\frac{x}{2}}\)
Notice that I have not changed the value of the expression; I just created a common denominator. Now, combine like terms.
\(\textcolor{red}{\frac{4x}{2}-\frac{1}{2}+\frac{x}{2}\\ \frac{5x}{2}-\frac{1}{2}\\ \frac{5x-1}{2}}\)
Therefore, \(\frac{x^2-5x+6}{2x-1+\frac{x}{2}}=\frac{x^2-5x+6}{\frac{5x-1}{2}}\) . Now, it is time to simplify.
\(\frac{x^2-5x+6}{\frac{5x-1}{2}}*\frac{2}{2}\\ \frac{2(x^2-5x+6)}{5x-1}\\ \frac{2(x-2)(x-3)}{5x-1}\)
Even after I factored the numerator completely, I did not find any common factors present in both the numerator and the denominator, so this expression is already in simplest form.