Simplify the Expression: Solving (2/3 + m/4)² - 4/3 - (m/4 - 2/3)²

To solve this problem, let's follow a detailed approach:

• First, expand both squares:
• $\left(\frac{2}{3} + \frac{m}{4}\right)^2 = \left(\frac{2}{3}\right)^2 + \frac{2 \times 2}{3 \times 4} \times m + \left(\frac{m}{4}\right)^2 = \frac{4}{9} + \frac{m}{6} + \frac{m^2}{16}$
• $\left(\frac{m}{4} - \frac{2}{3}\right)^2 = \left(\frac{m}{4}\right)^2 - \frac{2 \times m}{12} + \left(\frac{2}{3}\right)^2 = \frac{m^2}{16} - \frac{m}{6} + \frac{4}{9}$
• Now substitute and simplify into the expression:

• Expression becomes: $\left( \frac{4}{9} + \frac{m}{6} + \frac{m^2}{16} \right) - \frac{4}{3} - \left( \frac{m^2}{16} - \frac{m}{6} + \frac{4}{9} \right)$

• Observe that $\frac{m^2}{16} - \frac{m^2}{16}$ cancels out.
• Now simplify the remaining terms: $\frac{4}{9} + \frac{m}{6} - \frac{4}{3} + \frac{m}{6} - \frac{4}{9} = \frac{2m}{6} - \frac{4}{3}$
• Use common denominators to combine final terms:
• $\frac{2m}{6} = \frac{m}{3}$, $\frac{4}{3} = \frac{12}{9}$, resulting in: $\frac{m}{3} - \frac{12}{9}$
• Recognize that this is a difference of squares:
• $\frac{m}{3} - \frac{12}{9} = \frac{(\sqrt{2m} + 2)(\sqrt{2m} - 2)}{3}$

Therefore, the simplified expression is given by the choice: $\frac{(\sqrt{2m}+2)(\sqrt{2m}-2)}{3}$.