cyrsvab

2022-02-04

How do you multiply $\left({m}^{2}-4\mp +{p}^{2}\right)\left({m}^{2}+4\mp -{p}^{2}\right)$?

Explanation:
$\left({m}^{2}-4\mp +{p}^{2}\right)\left({m}^{2}+4\mp -{p}^{2}\right)$
$={m}^{2}\left({m}^{2}+4\mp -{p}^{2}\right)-4\mp \left({m}^{2}+4\mp -{p}^{2}\right)+{p}^{2}\left({m}^{2}+4\mp -{p}^{2}\right)$
$={m}^{4}+4{m}^{3}p-{p}^{2}{m}^{2}-4{m}^{3}p-16{m}^{2}{p}^{2}+4{\mp }^{3}+{p}^{2}{m}^{2}+4{\mp }^{3}-{p}^{4}$
$={m}^{4}-16{m}^{2}{p}^{2}+8{\mp }^{3}-{p}^{4}$

stefjumnmt

One can use formula $\left(a-b\right)\left(a+b\right)={a}^{2}-{b}^{2}$ as follows:
$\left({m}^{2}-4\mp +{p}^{2}\right)\left({m}^{2}+4\mp -{p}^{2}\right)$
$=\left({m}^{2}-\left(4\mp -{p}^{2}\right)\right)\left({m}^{2}+\left(4\mp -{p}^{2}\right)\right)\right)$
$={\left({m}^{2}\right)}^{2}-{\left(4\mp -{p}^{2}\right)}^{2}$
$={m}^{4}-\left(16{m}^{2}{p}^{2}-8{\mp }^{3}+{p}^{4}\right)$
$={m}^{4}-16{m}^{2}{p}^{2}+8{\mp }^{3}-{p}^{4}$

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