To simplify the given expression, we perform the indicated operation and leave the answer in factored form.
Given:
[tex]\[
\frac{6}{x-3} - \frac{3}{x+7}
\][/tex]
1. Find a common denominator:
The denominators of the fractions are different, so we'll find a common denominator. The common denominator will be the product of both denominators:
[tex]\[
(x - 3)(x + 7)
\][/tex]
2. Rewrite each fraction with the common denominator:
[tex]\[
\frac{6}{x-3} = \frac{6(x+7)}{(x-3)(x+7)}
\][/tex]
[tex]\[
\frac{3}{x+7} = \frac{3(x-3)}{(x-3)(x+7)}
\][/tex]
3. Combine the fractions:
[tex]\[
\frac{6(x+7)}{(x-3)(x+7)} - \frac{3(x-3)}{(x-3)(x+7)} = \frac{6(x+7) - 3(x-3)}{(x-3)(x+7)}
\][/tex]
4. Simplify the numerator:
[tex]\[
6(x + 7) - 3(x - 3) = 6x + 42 - 3x + 9 = 3x + 51
\][/tex]
So the combined and simplified expression is:
[tex]\[
\frac{3x + 51}{(x-3)(x+7)}
\][/tex]
5. Factor the numerator:
Notice that [tex]\(3x + 51\)[/tex] can be factored as [tex]\(3(x + 17)\)[/tex]:
[tex]\[
3(x + 17)
\][/tex]
6. Write the final expression in factored form:
So, the simplified and factored form of the expression is:
[tex]\[
\boxed{\frac{3(x + 17)}{(x - 3)(x + 7)}}
\][/tex]
