To determine which expression is equivalent to the given quotient [tex]\(\frac{\frac{3y^2 - 3}{x^2 + 3}}{\frac{x + 1}{x(x+3)}}\)[/tex], we need to simplify this complex fraction step-by-step.
### Step 1: Simplify the Complex Fraction
First, rewrite the given complex fraction:
[tex]\[
\frac{\frac{3y^2 - 3}{x^2 + 3}}{\frac{x + 1}{x(x + 3)}}
\][/tex]
Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Thus:
[tex]\[
\frac{\frac{3y^2 - 3}{x^2 + 3}}{\frac{x + 1}{x(x + 3)}} = \frac{3y^2 - 3}{x^2 + 3} \cdot \frac{x(x + 3)}{x + 1}
\][/tex]
### Step 2: Multiplication of Fractions
Now perform the multiplication:
[tex]\[
\frac{3y^2 - 3}{x^2 + 3} \cdot \frac{x(x + 3)}{x + 1}
\][/tex]
### Step 3: Factor and Simplify
First, factor the numerator and the denominator where possible:
1. The numerator of the first fraction [tex]\(3y^2 - 3\)[/tex] can be factored:
[tex]\[
3y^2 - 3 = 3(y^2 - 1) = 3(y - 1)(y + 1)
\][/tex]
2. The denominator of the first fraction [tex]\(x^2 + 3\)[/tex] cannot be factored further.
3. The numerator of the second fraction [tex]\(x(x + 3)\)[/tex] is already factored.
4. The denominator of the second fraction [tex]\(x + 1\)[/tex] is already factored.
Thus, the expression can be rewritten as:
[tex]\[
\frac{3(y - 1)(y + 1)}{x^2 + 3} \cdot \frac{x(x + 3)}{x + 1}
\][/tex]
### Step 4: Simplify by Multiplication
Combine the fractions:
[tex]\[
\frac{3(y - 1)(y + 1) \cdot x(x + 3)}{(x^2 + 3)(x + 1)}
\][/tex]
### Step 5: Identify Equivalent Expressions
Now we need to compare this simplified expression with the given choices [tex]\(A\)[/tex], [tex]\(B\)[/tex], [tex]\(C\)[/tex], and [tex]\(D\)[/tex].
None of the given options simplifies directly to match the structure of our expression.
Therefore, after evaluating all steps of simplification meticulously, we conclude that none of the options provided match the simplified form. Therefore, the correct response is:
[tex]\[
\boxed{\text{None of the above}}
\][/tex]
