Factor (n^2)/9+2mn+9m^2

$\frac{n^{2}}{9} + 2 m n + 9 m^{2}$

Rewrite $9$ as $3^{2}$ .

$\frac{n^{2}}{3^{2}} + 2 m n + 9 m^{2}$

Rewrite $\frac{n^{2}}{3^{2}}$ as ${(\frac{n}{3})}^{2}$ .

${(\frac{n}{3})}^{2} + 2 m n + 9 m^{2}$

Rewrite $9 m^{2}$ as ${(3 m)}^{2}$ .

${(\frac{n}{3})}^{2} + 2 m n + {(3 m)}^{2}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 m n = 2 \cdot \frac{n}{3} \cdot (3 m)$

Rewrite the polynomial.

${(\frac{n}{3})}^{2} + 2 \cdot \frac{n}{3} \cdot (3 m) + {(3 m)}^{2}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = \frac{n}{3}$ and $b = 3 m$ .

${(\frac{n}{3} + 3 m)}^{2}$