To simplify the given expression:
[tex]\[
\frac{4w}{w-2} + \frac{3w}{w-3}
\][/tex]
we need to find a common denominator and combine the fractions. The common denominator of the two fractions is [tex]\((w - 2)(w - 3)\)[/tex]. First, we rewrite each fraction with the common denominator:
[tex]\[
\frac{4w}{w-2} = \frac{4w(w-3)}{(w-2)(w-3)}
\][/tex]
and
[tex]\[
\frac{3w}{w-3} = \frac{3w(w-2)}{(w-2)(w-3)}
\][/tex]
Next, we combine the two fractions:
[tex]\[
\frac{4w(w-3) + 3w(w-2)}{(w-2)(w-3)}
\][/tex]
We will now distribute within the numerators:
[tex]\[
4w(w-3) = 4w^2 - 12w
\][/tex]
[tex]\[
3w(w-2) = 3w^2 - 6w
\][/tex]
Adding these together:
[tex]\[
4w^2 - 12w + 3w^2 - 6w = 7w^2 - 18w
\][/tex]
So, combining these into one fraction, we have:
[tex]\[
\frac{7w^2 - 18w}{(w-2)(w-3)}
\][/tex]
The denominator can be factored as follows:
[tex]\[
(w-2)(w-3) = w^2 - 5w + 6
\][/tex]
Thus, the combined fraction is:
[tex]\[
\frac{7w^2 - 18w}{w^2 - 5w + 6}
\][/tex]
Among the provided choices:
A. [tex]\(\frac{7w^2 - 5}{w^2 - 5w + 6}\)[/tex]
B. [tex]\(\frac{7w}{w^2 + 6}\)[/tex]
C. [tex]\(\frac{7w^2 - 18w}{w^2 - 5w + 6}\)[/tex]
D. [tex]\(\frac{7 \psi}{2w - 5}\)[/tex]
Option C, [tex]\(\frac{7w^2 - 18w}{w^2 - 5w + 6}\)[/tex], matches our result.
Therefore, the correct answer is:
C. [tex]\(\frac{7w^2 - 18w}{w^2 - 5w + 6}\)[/tex]
