Question
Multiply the binomials: $\left(\frac{3}{4} a^{2}+3 b^{2}\right)$ and $ 4\left(a^{2}-\frac{2}{3} b^{2}\right)$
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Answer
$\left(\frac{3}{4} a^{2}+3 b^{2}\right) \times 4\left(a^{2}-\frac{2}{3} b^{2}\right)$
$=\left(\frac{3}{4} a^{2}+3 b^{2}\right) \times\left(4 a^{2}-\frac{8}{3} b^{2}\right)$
$=\frac{3}{4} a^{2} \times\left(4 a^{2}-\frac{8}{3} b^{2}\right)+3 b^{2} \times\left(4 a^{2}-\frac{8}{3} b^{2}\right)$
$=\frac{3}{4} a^{2} \times 4 a^{2}-\frac{3}{4} a^{2} \times \frac{8}{3} b^{2}+3 b^{2} \times 4 a^{2}-3 b^{2} \times \frac{8}{3} b^{2}$
$=3 a^{4}-2 a^{2} b^{2}+12 a^{2} b^{2}-8 b^{4}$
$=3 a^{4}+10 a^{2} b^{2}-8 b^{4}$
$=\left(\frac{3}{4} a^{2}+3 b^{2}\right) \times\left(4 a^{2}-\frac{8}{3} b^{2}\right)$
$=\frac{3}{4} a^{2} \times\left(4 a^{2}-\frac{8}{3} b^{2}\right)+3 b^{2} \times\left(4 a^{2}-\frac{8}{3} b^{2}\right)$
$=\frac{3}{4} a^{2} \times 4 a^{2}-\frac{3}{4} a^{2} \times \frac{8}{3} b^{2}+3 b^{2} \times 4 a^{2}-3 b^{2} \times \frac{8}{3} b^{2}$
$=3 a^{4}-2 a^{2} b^{2}+12 a^{2} b^{2}-8 b^{4}$
$=3 a^{4}+10 a^{2} b^{2}-8 b^{4}$
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