Ignore $m$ and $n$ for a moment and just look at 12 and 18. What is the greatest integer that divides both 12 and 18?

Compare $m^3$ with $m^2$, and $n$ with $n^2$. For each variable, use the smaller exponent as the shared factor, then divide each term by the full GCF to see what goes inside the parentheses.

In a Desmos calculator line, type:

orig = 12*m^3*n + 18*m^2*n^2

This stores the original expression so you can compare it with each choice.

On new lines, type:

• exprA = 6*m^3*n*(2 + 3*n) (choice A)
• exprB = 6*m^2*n^2*(2*m + 3) (choice B)
• exprC = 12*m^2*n*(m + n) (choice C)
• exprD = 6*m^2*n*(2*m + 3*n) (choice D)

These represent the four options.

Choose simple values, such as m = 1 and n = 2, by typing them on separate lines. Desmos will compute numerical values for orig, exprA, exprB, exprC, and exprD. The equivalent expression will have the same value as orig. You can confirm by trying another set of values (for example, m = 2, n = -1) and checking which expression still matches orig.

You are looking for a common factor that multiplies both terms in $12m^3n + 18m^2n^2$.

That common factor (the GCF) will have:

• A numerical part that divides both 12 and 18.
• A variable part with powers of $m$ and $n$ that appear in both terms.

Once you know this GCF, you can factor it out in front of parentheses.

Look at the coefficients 12 and 18.

• Factors of 12: $1, 2, 3, 4, 6, 12$
• Factors of 18: $1, 2, 3, 6, 9, 18$

The greatest common factor of 12 and 18 is $6$.

Now look at the variables in each term:

• First term: $12m^3n$ has $m^3$ and $n^1$.
• Second term: $18m^2n^2$ has $m^2$ and $n^2$.

For each variable, the common factor uses the smaller exponent:

• For $m^3$ and $m^2$, the smaller exponent is 2, so the common factor in $m$ is $m^2$.
• For $n^1$ and $n^2$, the smaller exponent is 1, so the common factor in $n$ is $n$.

So the variable part of the GCF is $m^2n$. Combined with the numerical part, the full GCF is $6m^2n$.

Factor $6m^2n$ out of each term by dividing:

• $12m^3n / 6m^2n = 2m$
• $18m^2n^2 / 6m^2n = 3n$

So

This matches choice D, so the equivalent expression is $6m^2n(2m + 3n)$.