$\displaystyle 4a^4m^2c^2=4a^2(b^2c^2-b^4+a^2m^2c^2-a^2m^2b^2)$ How can this be simplified to give $\displaystyle c^2= b^2+a^2m^2$?
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First of all simplify by $\displaystyle 4a^2$: $\displaystyle a^2m^2c^2=b^2c^2-b^4+a^2m^2c^2-a^2m^2b^2\Rightarrow b^2c^2=b^4+a^2m^2b^2$ Now, simplify by $\displaystyle b^2$ and you'll get the result.
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