To simplify the rational expression [tex]\(\frac{6x^2 - 54}{5x^2 + 15x}\)[/tex], we need to factor both the numerator and the denominator, and then simplify by cancelling common factors.
1. Factor the Numerator:
The numerator is [tex]\(6x^2 - 54\)[/tex].
Notice that [tex]\(6x^2 - 54\)[/tex] has a common factor of 6. So, we can factor out the 6:
[tex]\[
6x^2 - 54 = 6(x^2 - 9)
\][/tex]
Notice that [tex]\(x^2 - 9\)[/tex] is a difference of squares, which can be factored further:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
So, the numerator fully factored is:
[tex]\[
6(x - 3)(x + 3)
\][/tex]
2. Factor the Denominator:
The denominator is [tex]\(5x^2 + 15x\)[/tex].
Notice that [tex]\(5x^2 + 15x\)[/tex] has a common factor of [tex]\(5x\)[/tex]. So, we can factor out the [tex]\(5x\)[/tex]:
[tex]\[
5x^2 + 15x = 5x(x + 3)
\][/tex]
3. Write the Rational Expression with the Factors:
Now that we have factored both the numerator and the denominator, we can write:
[tex]\[
\frac{6x^2 - 54}{5x^2 + 15x} = \frac{6(x - 3)(x + 3)}{5x(x + 3)}
\][/tex]
4. Simplify by Cancelling Common Factors:
We notice that [tex]\((x + 3)\)[/tex] is a common factor in both the numerator and the denominator, so we can cancel [tex]\((x + 3)\)[/tex] from both:
[tex]\[
\frac{6(x - 3)\cancel{(x + 3)}}{5x\cancel{(x + 3)}} = \frac{6(x - 3)}{5x}
\][/tex]
So, the simplified form of the given rational expression is:
[tex]\[
\frac{6(x - 3)}{5x}
\][/tex]
Therefore, the correct answer is:
A. [tex]\(\frac{6(x - 3)}{5x}\)[/tex]
