select the correct answer. if no denominator equals zero, which expression is equivalent to $\frac{x +…

select the correct answer. if no denominator equals zero, which expression is equivalent to $\frac{x + 2}{x^{2}-16}div\frac{2x + 4}{x^{2}+3x - 4}$?\na. $\frac{x - 1}{2(x + 4)}$\nb. $\frac{x - 1}{2(x + 1)}$\nc. $\frac{x - 1}{2(x - 4)}$\nd. $\frac{(x + 2)^{2}}{2(x + 4)^{2}}$

select the correct answer. if no denominator equals zero, which expression is equivalent to $\frac{x + 2}{x^{2}-16}div\frac{2x + 4}{x^{2}+3x - 4}$?\na. $\frac{x - 1}{2(x + 4)}$\nb. $\frac{x - 1}{2(x + 1)}$\nc. $\frac{x - 1}{2(x - 4)}$\nd. $\frac{(x + 2)^{2}}{2(x + 4)^{2}}$

Answer

Explanation:

Step1: Rewrite division as multiplication

Dividing by a fraction is the same as multiplying by its reciprocal. So, $\frac{x + 2}{x^{2}-16}\div\frac{2x + 4}{x^{2}+3x - 4}=\frac{x + 2}{x^{2}-16}\times\frac{x^{2}+3x - 4}{2x + 4}$.

Step2: Factor the expressions

Factor $x^{2}-16=(x + 4)(x - 4)$ using the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$ where $a=x$ and $b = 4$. Factor $x^{2}+3x - 4=(x + 4)(x - 1)$ using the formula for factoring quadratic $ax^{2}+bx + c=a(x - x_1)(x - x_2)$ and $2x + 4=2(x + 2)$. The expression becomes $\frac{x + 2}{(x + 4)(x - 4)}\times\frac{(x + 4)(x - 1)}{2(x + 2)}$.

Step3: Cancel out common factors

Cancel out the common factors $(x + 2)$ and $(x + 4)$ in the numerator and denominator. We get $\frac{x - 1}{2(x - 4)}$.

Answer:

C. $\frac{x - 1}{2(x - 4)}$