Question 2: Factorize each of the following expresssions:
\((vi)\) \(\; \)\((x+1)(x-1)(x+2)(x-2)+13x^2 \)
Note: The question given in the book is incorrect. The correct form of the question should be: \((x+1)(x-1)(x+2)(x-2)+5x^2 \)
Solution:
$\displaystyle \ \ \ \ ( x+1)( x-1)( x+2)( x-2) +5x^{2}$
$\displaystyle =[( x+1)( x+2)][( x-1)( x-2)] +5x^{2}$
$\displaystyle =[ x( x+2) +1( x+2)][ x( x-2)\ -\ $$1( x-2)] +5x^{2}$
$\displaystyle =[ x^{2} +2x+x+2][ x^{2} -2x-x+2]\ +$$5x^{2}$
$\displaystyle =[ x^{2} +2+3x][ x^{2} +2-3x] +5x^{2}$
Let \(\; y=x^2+2\), we have
$\displaystyle =[ y+3x][ y-3x] +5x^{2}$
$\displaystyle =y( y-3x) +3x( y-3x) +5x^{2}$
$\displaystyle =y^{2} -3xy+3xy-9x^{2} +5x^{2}$
$\displaystyle =y^{2} -4x^{2}$
$\displaystyle =y^{2} -( 2x)^{2}$
$\displaystyle =( y+2x)( y-2x)$
$\displaystyle =\left( x^{2} +2+2x\right)\left( x^{2} +2-2x\right)$
$\displaystyle =\left( x^{2} +2x+2\right)\left( x^{2} -2x+2\right)$