To solve the expression [tex]\(\frac{3m-6}{4m+12} \cdot \frac{m^2+5m+6}{m^2-4}\)[/tex], let's go through the problem step by step.
### Step 1: Factorize where possible
Firstly, let's factorize each part of the given fractions:
1. Numerator of the first fraction: [tex]\(3m - 6\)[/tex]
- Common factor: [tex]\(3(m - 2)\)[/tex]
2. Denominator of the first fraction: [tex]\(4m + 12\)[/tex]
- Common factor: [tex]\(4(m + 3)\)[/tex]
3. Numerator of the second fraction: [tex]\(m^2 + 5m + 6\)[/tex]
- This is a quadratic expression. To factor it, we look for two numbers that multiply to [tex]\(6\)[/tex] (constant term) and add up to [tex]\(5\)[/tex] (coefficient of [tex]\(m\)[/tex]). The numbers [tex]\(2\)[/tex] and [tex]\(3\)[/tex] work.
[tex]\[
(m+2)(m+3)
\][/tex]
4. Denominator of the second fraction: [tex]\(m^2 - 4\)[/tex]
- This is a difference of squares.
[tex]\[
(m+2)(m-2)
\][/tex]
So, we can rewrite the fractions as follows:
[tex]\[
\frac{3(m-2)}{4(m+3)} \cdot \frac{(m+2)(m+3)}{(m+2)(m-2)}
\][/tex]
### Step 2: Simplify
Next, we simplify the expression by canceling out common factors in the numerator and the denominator.
The expression becomes:
[tex]\[
\frac{3(m-2) \cdot (m+2)(m+3)}{4(m+3) \cdot (m+2)(m-2)}
\][/tex]
We observe that [tex]\((m-2)\)[/tex], [tex]\((m+2)\)[/tex], and [tex]\((m+3)\)[/tex] appear in both the numerator and the denominator. These terms can be canceled out.
After canceling the common factors, the expression simplifies to:
[tex]\[
\frac{3}{4}
\][/tex]
Thus, the simplified result of the given operation is:
[tex]\[
\boxed{\frac{3}{4}}
\][/tex]
