Factor 4m^2n - 8mn^2 - 2mn.

Step-by-step solution

1. 1

   Find the greatest common factor coefficient by coefficient. The numerical coefficients are $4,-8,-2$, whose greatest common divisor is $2$.

2. 2

   Find the common variable powers. Every term contains $m$ to at least the first power and $n$ to at least the first power: the powers of $m$ are $2,1,1$ (minimum $1$) and of $n$ are $1,2,1$ (minimum $1$). So the full GCF is

   $\gcd=2mn$

3. 3

   Divide each term by 2mn, writing the quotients explicitly.

   $\frac{4m^2n}{2mn}=2m,\qquad \frac{-8mn^2}{2mn}=-4n,\qquad \frac{-2mn}{2mn}=-1$

   The third quotient is the step people get wrong: when a term is the GCF (up to sign), it does not vanish — it leaves behind $-1$.

4. 4

   Assemble the factored form.

   $4m^2n-8mn^2-2mn=2mn\left(2m-4n-1\right)$

5. 5

   Check whether the bracket factors further. $2m-4n-1$ is linear in both variables with no common factor (the $-1$ blocks pulling out a $2$), so the factoring is complete.

6. 6

   Verify by expanding and by random substitution. Distributing back: $2mn\cdot2m=4m^2n$, $2mn\cdot(-4n)=-8mn^2$, $2mn\cdot(-1)=-2mn$ ✓. Testing $60$ random pairs $(m,n)$ in $[-5,5]$ gives agreement to machine precision at every point ✓.