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Question Number 12257 by b.e.h.i.8.3.4.1.7@gmail.com last updated on 16/Apr/17

Answered by sma3l2996 last updated on 17/Apr/17

(a+b−2ab)^2 +(a+b−1)^2 =(a^2 +b^2 +4a^2 b^2 −4ab^2 +2a(b−2ab))+(a^2 +b^2 +1−2b+2a(b−1)) =2a^2 +2b^2 +4a^2 b^2 −4ab^2 +2ab−4a^2 b−2b+2ab−2a+1 =2a^2 (1+2b^2 −2b)+2a(2b−2b^2 −1)+2b^2 −2b+1 =(2b^2 −2b+1)(2a^2 −2a+1) so it′s correct

$$\left({a}+{b}−\mathrm{2}{ab}\right)^{\mathrm{2}} +\left({a}+{b}−\mathrm{1}\right)^{\mathrm{2}} =\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +\mathrm{4}{a}^{\mathrm{2}} {b}^{\mathrm{2}} −\mathrm{4}{ab}^{\mathrm{2}} +\mathrm{2}{a}\left({b}−\mathrm{2}{ab}\right)\right)+\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +\mathrm{1}−\mathrm{2}{b}+\mathrm{2}{a}\left({b}−\mathrm{1}\right)\right) \\ $$$$=\mathrm{2}{a}^{\mathrm{2}} +\mathrm{2}{b}^{\mathrm{2}} +\mathrm{4}{a}^{\mathrm{2}} {b}^{\mathrm{2}} −\mathrm{4}{ab}^{\mathrm{2}} +\mathrm{2}{ab}−\mathrm{4}{a}^{\mathrm{2}} {b}−\mathrm{2}{b}+\mathrm{2}{ab}−\mathrm{2}{a}+\mathrm{1} \\ $$$$=\mathrm{2}{a}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{2}{b}^{\mathrm{2}} −\mathrm{2}{b}\right)+\mathrm{2}{a}\left(\mathrm{2}{b}−\mathrm{2}{b}^{\mathrm{2}} −\mathrm{1}\right)+\mathrm{2}{b}^{\mathrm{2}} −\mathrm{2}{b}+\mathrm{1} \\ $$$$=\left(\mathrm{2}{b}^{\mathrm{2}} −\mathrm{2}{b}+\mathrm{1}\right)\left(\mathrm{2}{a}^{\mathrm{2}} −\mathrm{2}{a}+\mathrm{1}\right) \\ $$$${so}\:{it}'{s}\:{correct} \\ $$

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