Let's simplify the given rational expression step by step.
The rational expression given is:
[tex]\[
\frac{6 x^2 - 54}{5 x^2 + 15 x}
\][/tex]
First, let's factorize the numerator and the denominator separately.
### Step 1: Factorize the numerator
The numerator is [tex]\(6 x^2 - 54\)[/tex].
We can factor out the greatest common factor, which is [tex]\(6\)[/tex], from the numerator:
[tex]\[
6 x^2 - 54 = 6 (x^2 - 9)
\][/tex]
Notice that [tex]\(x^2 - 9\)[/tex] is a difference of squares, which can be further factored:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
Therefore, the factored form of the numerator is:
[tex]\[
6 (x^2 - 9) = 6 (x - 3)(x + 3)
\][/tex]
### Step 2: Factorize the denominator
The denominator is [tex]\(5 x^2 + 15 x\)[/tex].
We can factor out the greatest common factor, which is [tex]\(5 x\)[/tex], from the denominator:
[tex]\[
5 x^2 + 15 x = 5 x (x + 3)
\][/tex]
### Step 3: Write the rational expression with the factored numerator and denominator
Using the factored forms from Steps 1 and 2, the rational expression becomes:
[tex]\[
\frac{6 (x - 3)(x + 3)}{5 x (x + 3)}
\][/tex]
### Step 4: Simplify the rational expression
We can cancel out the common factors in the numerator and denominator. Specifically, [tex]\((x + 3)\)[/tex] appears in both the numerator and the denominator:
[tex]\[
\frac{6 (x - 3) \cancel{(x + 3)}}{5 x \cancel{(x + 3)}} = \frac{6 (x - 3)}{5 x}
\][/tex]
Thus, the simplified form of the rational expression is:
[tex]\[
\frac{6 (x - 3)}{5 x}
\][/tex]
### Conclusion
The simplified form of the rational expression [tex]\(\frac{6 x^2 - 54}{5 x^2 + 15 x}\)[/tex] is:
[tex]\[
\boxed{\frac{6(x-3)}{5 x}}
\][/tex]
So, the correct answer is:
B. [tex]\(\frac{6(x-3)}{5 x}\)[/tex]
