To simplify the given expression [tex]\(\frac{1}{x^2-5x+6} + \frac{2}{4x-x^2-3}\)[/tex], let's break it down step-by-step and combine the fractions appropriately.
Step 1: Factorize the denominators of both fractions.
- For [tex]\(x^2 - 5x + 6\)[/tex]:
[tex]\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\][/tex]
- For [tex]\(4x - x^2 - 3\)[/tex]:
[tex]\[
4x - x^2 - 3 = - (x^2 - 4x + 3) = - ((x-1)(x-3))
\][/tex]
Now the expression looks like:
[tex]\[
\frac{1}{(x-2)(x-3)} + \frac{2}{-(x-1)(x-3)}
\][/tex]
Step 2: Adjust the signs to combine under a common denominator:
[tex]\[
\frac{1}{(x-2)(x-3)} - \frac{2}{(x-1)(x-3)}
\][/tex]
Step 3: Determine the common denominator, which is [tex]\((x-2)(x-3)(x-1)\)[/tex]:
[tex]\[
\frac{(x-1) \cdot 1 - 2 \cdot (x-2)}{(x-2)(x-3)(x-1)}
\][/tex]
Step 4: Simplify the numerator:
[tex]\[
(x-1) - 2(x-2) = x - 1 - 2x + 4 = -x + 3
\][/tex]
So, the expression becomes:
[tex]\[
\frac{-x + 3}{(x-2)(x-3)(x-1)}
\][/tex]
Step 5: Notice that [tex]\(-x + 3\)[/tex] can be written as [tex]\(-(x - 3)\)[/tex], hence:
[tex]\[
\frac{-(x - 3)}{(x-2)(x-3)(x-1)}
\][/tex]
Cancelling the common [tex]\((x - 3)\)[/tex] term in the numerator and denominator, we get:
[tex]\[
\frac{-1}{(x-2)(x-1)}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{-1}{(x^2 - 3x + 2)}}
\][/tex]
This is the simplified form of the given expression as required.
