Question
Find the following product:
$\frac{2}{3}\text{abc}(\text{a}^2+\text{b}^2-3\text{c}^2)$
$\frac{2}{3}\text{abc}(\text{a}^2+\text{b}^2-3\text{c}^2)$
✓
Answer
$=\frac{2}{3}\text{abc}\times \text{a}^2+\frac{2}{3}\text{abc}\times \text{b}^2-\frac{2}{3}\text{abc}\times 3\text{c}^2$
$=\frac{2}{3}\text{a}\times \text{a}^2\times \text{b}\times \text{c}+\frac{2}{3}\text{a}\times \text{b}\times \text{b}^2\times \text{c}\\-\frac{2}{3}\times 3\times \text{a}\times \text{b}\times \text{c}\times \text{c}^2$
$=\frac{2}{3}\times \text{a}^{1+2}\times \text{b}\times \text{c}+\frac{2}{3}\times \text{a}\times \text{b}^{1+2}\times \text{c}\\-2\times \text{a}\times \text{b}\times \text{c}^{1+2}$
$=\frac{2}{3}\text{a}^3\text{bc}+\frac{2}{3}\text{ab}^3\text{c}-2\text{abc}^3$
$=\frac{2}{3}\text{a}\times \text{a}^2\times \text{b}\times \text{c}+\frac{2}{3}\text{a}\times \text{b}\times \text{b}^2\times \text{c}\\-\frac{2}{3}\times 3\times \text{a}\times \text{b}\times \text{c}\times \text{c}^2$
$=\frac{2}{3}\times \text{a}^{1+2}\times \text{b}\times \text{c}+\frac{2}{3}\times \text{a}\times \text{b}^{1+2}\times \text{c}\\-2\times \text{a}\times \text{b}\times \text{c}^{1+2}$
$=\frac{2}{3}\text{a}^3\text{bc}+\frac{2}{3}\text{ab}^3\text{c}-2\text{abc}^3$
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