Which expression is equivalent to [tex]\frac{m-4}{m+4} \div (m+2)[/tex]?

A. [tex]\frac{m-4}{(m+4)(m+2)}[/tex]
B. [tex]\frac{(m+4)(m+2)}{m-4}[/tex]
C. [tex]\frac{(m-4)(m+2)}{m+4}[/tex]
D. [tex]\frac{m+4}{(m-4)(m+2)}[/tex]



Answer :

Let's simplify the given expression step-by-step to determine which option is equivalent to [tex]\(\frac{m-4}{m+4} \div(m+2)\)[/tex].

### Step 1: Rewrite the Division as Multiplication by the Reciprocal
The expression [tex]\(\frac{m-4}{m+4} \div(m+2)\)[/tex] can be rewritten using the property of division as multiplication by the reciprocal:
[tex]\[ \frac{m-4}{m+4} \div (m+2) = \frac{m-4}{m+4} \times \frac{1}{m+2} \][/tex]

### Step 2: Multiply the Expressions
Next, we multiply the two fractions:
[tex]\[ \frac{m-4}{m+4} \times \frac{1}{m+2} = \frac{m-4}{(m+4)(m+2)} \][/tex]

### Step 3: Simplify the Expression
Since there are no common factors in the numerator and the denominator that we can cancel out, the expression remains as:
[tex]\[ \frac{m-4}{(m+4)(m+2)} \][/tex]

### Conclusion
Thus, the equivalent expression to [tex]\(\frac{m-4}{m+4} \div(m+2)\)[/tex] is:
[tex]\[ \frac{m-4}{(m+4)(m+2)} \][/tex]

So, the correct answer is:
[tex]\[ \frac{m-4}{(m+4)(m+2)} \][/tex]