Which of the following is equivalent to `(2m+6)/4times(6m-36)/(3m+9)`?

Options

A. `(12m^2-216)/(12m+36)`
B. `(8m-30)/(3m+13)`
C. `(m-6)/4`
D. m-6

Answer

To simplify the expression \(\frac{2m + 6}{4} \times \frac{6m - 36}{3m + 9}\), we will follow these steps:

### Step 1: Factor the numerators and denominators
First, we will factor out common terms from the expressions in the numerator and denominator.

- For \(2m + 6\), we can factor out \(2\):
\[ 2m + 6 = 2(m + 3) \]

- For \(6m - 36\), we can factor out \(6\):
\[ 6m - 36 = 6(m - 6) \]

- For \(3m + 9\), we can factor out \(3\):
\[ 3m + 9 = 3(m + 3) \]

### Step 2: Rewrite the expression
Now, substituting these factored forms back into the expression, we have:
\[ \frac{2(m + 3)}{4} \times \frac{6(m - 6)}{3(m + 3)} \]

### Step 3: Simplify the fractions
Next, we can simplify the expression by canceling common factors:

1. The \(m + 3\) in the numerator and denominator cancels out:
\[ \frac{2}{4} \times \frac{6(m - 6)}{3} \]

2. Now simplify \(\frac{2}{4}\) to \(\frac{1}{2}\) and \(\frac{6}{3}\) to \(2\):
\[ \frac{1}{2} \times 2(m - 6) \]

3. The \(2\) in the numerator and denominator cancels out:
\[ 1(m - 6) = m - 6 \]

### Final Result
Thus, the simplified expression is:
\[ m - 6 \]

### Conclusion
The equivalent expression is \(m - 6\).

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