To determine the equivalent expression for [tex]\(\frac{x+3}{x^2-2 x-3} \div \frac{x^2+2 x-3}{x+1}\)[/tex], we will follow these step-by-step procedures:
1. Express the division as multiplication by the reciprocal:
[tex]\[
\frac{x+3}{x^2-2 x-3} \div \frac{x^2+2 x-3}{x+1} = \frac{x+3}{x^2-2 x-3} \times \frac{x+1}{x^2+2 x-3}
\][/tex]
2. Factorize the denominators and numerators when possible:
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]
Therefore, the expression becomes:
[tex]\[
\frac{x + 3}{(x - 3)(x + 1)} \times \frac{x + 1}{(x + 3)(x - 1)}
\][/tex]
3. Simplify the expression by canceling the common factors in the numerator and denominator:
We note that [tex]\(x + 3\)[/tex] is a common factor in the numerator and the denominator, so it cancels out:
[tex]\[
\frac{1}{(x - 3)} \times \frac{1}{(x - 1)} = \frac{1}{(x - 3)(x - 1)}
\][/tex]
Here, the factors left in the denominator are [tex]\(x - 3\)[/tex] and [tex]\(x - 1\)[/tex].
4. Combine the simplified expression:
[tex]\[
\frac{1}{(x - 3)(x - 1)}
\][/tex]
5. Rewrite the denominator:
By expanding the factored form of the denominator, we get:
[tex]\[
(x - 3)(x - 1) = x^2 - 4x + 3
\][/tex]
Thus, the simplified expression equivalent to [tex]\(\frac{x+3}{x^2-2 x-3} \div \frac{x^2+2 x-3}{x+1}\)[/tex] is:
[tex]\[
\frac{1}{x^2 - 4x + 3}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{B. \frac{1}{x^2 - 4x + 3}}
\][/tex]
