Which expression is equivalent to [tex]\frac{c^2-4}{c+3} \div \frac{c+2}{3\left(c^2-9\right)}[/tex]?

A. [tex]\frac{c+3}{c^2-4} \div \frac{c+2}{3\left(c^2-9\right)}[/tex]

B. [tex]\frac{c^2-4}{c+3} \div \frac{3\left(c^2-9\right)}{c+2}[/tex]

C. [tex]\frac{c^2-4}{c+3} \cdot \frac{3\left(c^2-9\right)}{c+2}[/tex]

D. [tex]\frac{c+3}{c^2-4} \cdot \frac{c+2}{3\left(c^2-9\right)}[/tex]



Answer :

To determine which expression is equivalent to [tex]\(\frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)}\)[/tex], we start by understanding that division of fractions is equivalent to multiplying by the reciprocal of the second fraction.

Given expression:
[tex]\[ \frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)} \][/tex]

First rewrite the division as a multiplication by the reciprocal:
[tex]\[ \frac{c^2-4}{c+3} \times \frac{3(c^2-9)}{c+2} \][/tex]

Next, we'll simplify the components of the expression.

1. Factorize [tex]\(c^2 - 4\)[/tex] and [tex]\(c^2 - 9\)[/tex] using the difference of squares:
[tex]\[ c^2 - 4 = (c + 2)(c - 2) \][/tex]
[tex]\[ c^2 - 9 = (c + 3)(c - 3) \][/tex]

2. Substitute the factored forms into the expression:
[tex]\[ \frac{(c + 2)(c - 2)}{c + 3} \times \frac{3((c + 3)(c - 3))}{c + 2} \][/tex]

3. Simplify the expression by canceling out the common terms:
[tex]\[ = \frac{(c + 2)(c - 2)}{c + 3} \times \frac{3(c + 3)(c - 3)}{c + 2} \][/tex]

Notice that [tex]\(c + 2\)[/tex] in the numerator and denominator can be canceled:
[tex]\[ = \frac{(c - 2)}{c + 3} \times 3(c + 3)(c - 3) \][/tex]

Now cancel [tex]\(c + 3\)[/tex] in the numerator and denominator:
[tex]\[ = \frac{c - 2}{1} \times 3(c - 3) \][/tex]

This simplifies further to:
[tex]\[ = 3(c - 2) / (c - 3) \][/tex]

Thus, the simplified expression is:
[tex]\[ 3 \cdot \frac{(c - 2)}{(c - 3)} \][/tex]

So, the expression equivalent to [tex]\(\frac{c^2-4}{c+3} \div \frac{c+2}{3(c^2-9)}\)[/tex] is:
[tex]\[ \boxed{\frac{c^2-4}{c+3} \cdot \frac{3 \left( c^2-9 \right)}{c+2}} \][/tex]

Rewrite this expression for clarity as:
[tex]\[ 3 \cdot \frac{(c-2)}{(c-3)} \][/tex]