question_answer

If \[{{(5{{x}^{2}}+14x+2)}^{2}}-{{(4{{x}^{2}}-5x+7)}^{2}}\]is divided by \[({{x}^{2}}+x+1),\] then quotient 'q' and remainder '/-' respectively, are ____.

A) \[({{x}^{2}}+19x-5),0\]

B) \[9({{x}^{2}}+19x-5),0\]

C) \[({{x}^{2}}+19x-5),1\]

D) \[9({{x}^{2}}+19x-5),1\]

Correct Answer: B

Solution :

Let \[f(x)={{(5{{x}^{2}}+14x+2)}^{2}}-{{(4{{x}^{2}}-5x+7)}^{2}}\] \[=25{{x}^{4}}+196{{x}^{2}}+4+140{{x}^{3}}+56x+20{{x}^{2}}\] \[-16{{x}^{4}}-25{{x}^{2}}-49+40{{x}^{3}}+70x-56{{x}^{2}}\] \[=9{{x}^{4}}+180{{x}^{3}}+135{{x}^{2}}+126x-45\] and \[g(x)={{x}^{2}}+x+1\] By long division method, we have \[\therefore \]\[r=0\]and \[q=9{{x}^{2}}+171x-45\] i.e., \[9({{x}^{2}}+19x-5).\]