Simplify the Following Product
\[ \left(\frac{x}{2} – \frac{2}{5}\right) \left(\frac{2}{5} – \frac{x}{2}\right) – x^2 + 2x \]
Solution:
\[ = -\left(\frac{x}{2} – \frac{2}{5}\right)^2 – x^2 + 2x \]
\[ = -\left[ \left(\frac{x}{2}\right)^2 -2\left(\frac{x}{2}\right)\left(\frac{2}{5}\right) +\left(\frac{2}{5}\right)^2 \right] – x^2 + 2x \]
\[ = -\left( \frac{x^2}{4} -\frac{2x}{5} +\frac{4}{25} \right) – x^2 + 2x \]
\[ = -\frac{x^2}{4} +\frac{2x}{5} -\frac{4}{25} – x^2 + 2x \]
\[ = -\frac{5x^2}{4} +\frac{12x}{5} -\frac{4}{25} \]