Let's subtract the rational expressions and simplify the result.
We start with the given expressions:
[tex]\[
\frac{-3 x^2}{x-3} - \frac{9-4x^2}{x-3}
\][/tex]
1. Combine the expressions:
Since both fractions have the same denominator ([tex]\(x-3\)[/tex]), we can combine them by subtracting the numerators directly:
[tex]\[
\frac{-3 x^2}{x-3} - \frac{9-4x^2}{x-3} = \frac{-3 x^2 - (9-4x^2)}{x-3}
\][/tex]
2. Distribute the subtraction across the numerator:
[tex]\[
\frac{-3 x^2 - 9 + 4 x^2}{x-3}
\][/tex]
3. Combine like terms:
Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
\frac{(-3 x^2 + 4 x^2) - 9}{x-3}
\][/tex]
[tex]\[
\frac{x^2 - 9}{x-3}
\][/tex]
4. Factor the numerator:
Notice that [tex]\(x^2 - 9\)[/tex] is a difference of squares, which can be factored as [tex]\((x + 3)(x - 3)\)[/tex]:
[tex]\[
\frac{(x + 3)(x - 3)}{x-3}
\][/tex]
5. Simplify the expression:
The [tex]\(x-3\)[/tex] terms cancel out:
[tex]\[
\frac{(x + 3)(x - 3)}{x-3} = x + 3
\][/tex]
Therefore, the simplified result is:
[tex]\[
\boxed{x + 3}
\][/tex]
