To determine which expression is equivalent to [tex]\(\frac{x+3}{x^2 - 2x - 3} \div \frac{x^2 + 2x - 3}{x + 1}\)[/tex], we need to follow these steps:
### Step 1: Simplify the Division of Fractions
Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Thus the given expression can be rewritten as:
[tex]\[
\frac{x+3}{x^2 - 2x - 3} \div \frac{x^2 + 2x - 3}{x+1} = \frac{x+3}{x^2 - 2x - 3} \times \frac{x+1}{x^2 + 2x - 3}
\][/tex]
### Step 2: Examine the Numerator and Denominator
We have two parts to consider now:
1. [tex]\(\frac{x + 3}{x^2 - 2x - 3}\)[/tex]
2. [tex]\(\frac{x + 1}{x^2 + 2x - 3}\)[/tex]
### Step 3: Factorize the Denominators
To make our simplifications easier, we factorize the denominators.
For [tex]\(x^2 - 2x - 3\)[/tex]:
[tex]\[
x^2 - 2x - 3 = (x - 3)(x + 1)
\][/tex]
For [tex]\(x^2 + 2x - 3\)[/tex]:
[tex]\[
x^2 + 2x - 3 = (x + 3)(x - 1)
\][/tex]
### Step 4: Substitute the Factored Forms
Replace the factored forms in our original expression:
[tex]\[
\frac{x + 3}{(x - 3)(x + 1)} \times \frac{x + 1}{(x + 3)(x - 1)}
\][/tex]
### Step 5: Simplify the Expression
Now, simplify the expression by canceling common terms:
[tex]\[
\frac{x + 3}{(x - 3)(x + 1)} \times \frac{x + 1}{(x + 3)(x - 1)} = \frac{(x + 3) \cdot (x + 1)}{(x - 3)(x + 1) \cdot (x + 3)(x - 1)}
\][/tex]
[tex]\[
= \frac{1}{(x - 3)(x - 1)}
\][/tex]
### Step 6: Determine the Equivalent Expression
Now, we compare our simplified expression [tex]\(\frac{1}{(x - 3)(x - 1)}\)[/tex] with the given answer choices:
[tex]\[
(x - 3)(x - 1) = x^2 - 4x + 3
\][/tex]
Therefore:
[tex]\[
\frac{1}{(x - 3)(x - 1)} = \frac{1}{x^2 - 4x + 3}
\][/tex]
Thus, the answer choice that matches [tex]\(\frac{1}{x^2 - 4x + 3}\)[/tex] is:
### Final Answer
[tex]\[
\boxed{B}
\][/tex]
