To determine which expression is equivalent to the given quotient
[tex]\[
\frac{\frac{3x^2 - 3}{x^2 + 3x}}{\frac{x + 1}{x + 3}},
\][/tex]
we need to simplify it step-by-step.
1. Simplify the numerator:
[tex]\[
\frac{3x^2 - 3}{x^2 + 3x}
\][/tex]
Factor out the common term in the numerator and denominator:
[tex]\[
\frac{3(x^2 - 1)}{x(x + 3)}
\][/tex]
Further factorize [tex]\(x^2 - 1\)[/tex] using the difference of squares:
[tex]\[
\frac{3(x - 1)(x + 1)}{x(x + 3)}
\][/tex]
2. Simplify the denominator:
[tex]\[
\frac{x + 1}{x + 3}
\][/tex]
3. Rewrite the entire expression:
[tex]\[
\frac{\frac{3(x - 1)(x + 1)}{x(x + 3)}}{\frac{x + 1}{x + 3}}
\][/tex]
4. Invert the denominator and multiply:
[tex]\[
\frac{3(x - 1)(x + 1)}{x(x + 3)} \times \frac{x + 3}{x + 1}
\][/tex]
The [tex]\((x + 3)\)[/tex] terms and [tex]\((x + 1)\)[/tex] terms cancel out:
[tex]\[
\frac{3(x - 1) \cancel{(x + 1)}}{x \cancel{(x + 3)}} \times \frac{\cancel{(x + 3)}}{\cancel{(x + 1)}} = \frac{3(x - 1)}{x}
\][/tex]
Simplify the expression further by observing that this can be written as:
[tex]\[
\frac{3(x - 1)}{x} = 3 \left( \frac{x - 1}{x} \right)
\][/tex]
Given the choices:
A. [tex]\(3(x-1)\)[/tex]
B. [tex]\(\frac{-(x+3)}{x-1}\)[/tex]
C. [tex]\(3x(x-1)\)[/tex]
D. [tex]\(\frac{-3(x-1)}{x+3}\)[/tex]
Option B is the correct answer:
[tex]\[
\boxed{B}
\][/tex]
