Simplify the Expression: m²/9 - 4mn/3 + 4n² Step-by-Step

$\frac{m^2}{9}-\frac{4}{3}mn+4n^2=\text{?}$

To solve this problem, we'll simplify the expression $\frac{m^2}{9} - \frac{4}{3}mn + 4n^2$ to reveal a possible square of a binomial:

• Step 1: Recognize the expression $\frac{m^2}{9} - \frac{4}{3}mn + 4n^2$ may have terms fitting the identity $(a - b)^2 = a^2 - 2ab + b^2$.
• Step 2: Identify parts of the expression:
  - $a^2 = \frac{m^2}{9}$, so $a = \frac{m}{3}$.
  - $-2ab = -\frac{4}{3}mn$. Since $a = \frac{m}{3}$, solve for $b$:
  $2 \times \frac{m}{3} \times b = \frac{4}{3}mn \Rightarrow 2b = 4n \Rightarrow b = 2n$.
• Step 3: Check $b^2$: $b^2 = (2n)^2 = 4n^2$, which matches the $4n^2$ term in the expression.

Therefore, the correct form of the expression is:
$(\frac{m}{3} - 2n)^2$.

Thus, the expression simplifies to:

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