select the correct answer. which expression is equivalent to $\frac{x + 3}{x^{2}-2x - 3}div\frac{x^{2}+2x…

select the correct answer. which expression is equivalent to $\frac{x + 3}{x^{2}-2x - 3}div\frac{x^{2}+2x - 3}{x + 1}$ if no denominator equals zero?\na. $\frac{1}{x^{2}-2x - 3}$\nb. $\frac{1}{x^{2}-4x + 3}$\nc. $\frac{1}{x^{2}+2x - 3}$\nd. $\frac{x + 3}{x + 1}$

select the correct answer. which expression is equivalent to $\frac{x + 3}{x^{2}-2x - 3}div\frac{x^{2}+2x - 3}{x + 1}$ if no denominator equals zero?\na. $\frac{1}{x^{2}-2x - 3}$\nb. $\frac{1}{x^{2}-4x + 3}$\nc. $\frac{1}{x^{2}+2x - 3}$\nd. $\frac{x + 3}{x + 1}$

Answer

Answer:

B. $\frac{1}{x^{2}-4x + 3}$

Explanation:

Step1: Rewrite division as multiplication

$\frac{x + 3}{x^{2}-2x - 3}\div\frac{x^{2}+2x - 3}{x + 1}=\frac{x + 3}{x^{2}-2x - 3}\times\frac{x + 1}{x^{2}+2x - 3}$

Step2: Factor the quadratic expressions

$x^{2}-2x - 3=(x - 3)(x+1)$ and $x^{2}+2x - 3=(x + 3)(x - 1)$ So the expression becomes $\frac{x + 3}{(x - 3)(x + 1)}\times\frac{x + 1}{(x + 3)(x - 1)}$

Step3: Cancel out common factors

Cancel out $(x + 3)$ and $(x + 1)$ terms. We get $\frac{1}{(x - 3)(x - 1)}=\frac{1}{x^{2}-4x + 3}$