Question

what is the quotient?
\\(\frac{2m + 4}{8} div \frac{m + 2}{6}\\)
\\(\bigcirc \frac{24}{(m + 2)^2}\\)
\\(\bigcirc \frac{(m + 2)^2}{24}\\)
\\(\bigcirc \frac{2}{3}\\)
\\(\bigcirc \frac{3}{2}\\)

Explanation:

Step1: Rewrite division as multiplication

To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. So, \(\frac{2m + 4}{8}\div\frac{m + 2}{6}=\frac{2m + 4}{8}\times\frac{6}{m + 2}\).

Step2: Factor the numerator

Factor out 2 from the numerator \(2m + 4\), we get \(2(m + 2)\). So the expression becomes \(\frac{2(m + 2)}{8}\times\frac{6}{m + 2}\).

Step3: Simplify the fractions

Simplify \(\frac{2}{8}\) to \(\frac{1}{4}\), then we have \(\frac{1(m + 2)}{4}\times\frac{6}{m + 2}\). The \((m + 2)\) terms cancel out, and we multiply the remaining numbers: \(\frac{6}{4}=\frac{3}{2}\)? Wait, no, wait. Wait, \(\frac{2(m + 2)}{8}\times\frac{6}{m + 2}=\frac{2\times6\times(m + 2)}{8\times(m + 2)}\). Cancel \((m + 2)\) from numerator and denominator, then we have \(\frac{12}{8}=\frac{3}{2}\)? Wait, no, 26=12, 8 remains. Wait, 12/8 simplifies to 3/2? Wait, no, 2(m+2)6 = 12(m+2), 8*(m+2)=8(m+2). Then cancel (m+2), we get 12/8 = 3/2? Wait, but let's check again. Wait, 2m +4 is 2(m +2), so:

\(\frac{2(m + 2)}{8}\times\frac{6}{m + 2}=\frac{2\times6}{8}=\frac{12}{8}=\frac{3}{2}\)? Wait, no, that's not right. Wait, 2*6=12, 8 is the denominator. 12 divided by 8 is 3/2? Wait, but let's do it step by step.

Wait, \(\frac{2(m + 2)}{8}\times\frac{6}{m + 2}\). The \((m + 2)\) in the numerator and denominator cancels. Then we have \(\frac{2\times6}{8}=\frac{12}{8}=\frac{3}{2}\)? Wait, but that's not one of the options? Wait, no, wait, I think I made a mistake. Wait, the original problem is \(\frac{2m + 4}{8}\div\frac{m + 2}{6}\). Let's re - do it.

Wait, \(\frac{2m + 4}{8}\div\frac{m + 2}{6}=\frac{2m+4}{8}\times\frac{6}{m + 2}\). Factor 2m + 4 as 2(m + 2). So \(\frac{2(m + 2)}{8}\times\frac{6}{m + 2}\). Now, multiply the numerators: 2(m + 2)6 = 12(m + 2). Multiply the denominators: 8(m + 2)=8(m + 2). Now, cancel (m + 2) from numerator and denominator: \(\frac{12}{8}=\frac{3}{2}\)? Wait, but the options have 3/2 as an option? Wait, the last option is 3/2. Wait, but let's check again.

Wait, 2(m + 2) 6 = 12(m + 2), 8(m + 2)=8(m + 2). Cancel (m + 2), we get 12/8 = 3/2. Yes, that's correct. So the quotient is 3/2.

Answer:

\(\frac{3}{2}\) (the last option, which is \(\boldsymbol{\frac{3}{2}}\))