Question 2: Factorize each of the following expresssions:
\((v)\) \(\; \)\((x+1)(x+2)(x+3)(x+6)-3x^2 \)
Solution:
$\displaystyle\ \ \ \ ( x+1)( x+2)( x+3)( x+6) -3x^{2}$
$\displaystyle =[( x+1)( x+6)][( x+2)( x+3)] -3x^{2}$
$\displaystyle =[ x( x+6) +1( x+6)][ x( x+3)\ +$$2( x+3)] -3x^{2}$
$\displaystyle =[ x^{2} +6x+x+6][ x^{2} +3x+2x+6]\ -$$3x^{2}$
$\displaystyle =[ x^{2} +6+7x][ x^{2} +6+5x] -3x^{2}$
Let \(\; y=x^2+6\), we have
$\displaystyle =[ y+7x][ y+5x] -3x^{2}$
$\displaystyle =y( y+5x) +7x( y+5x) -3x^{2}$
$\displaystyle =y^{2} +5xy+7xy+35x^{2} -3x^{2}$
$\displaystyle =y^{2} +12xy+32x^{2}$
$\displaystyle =y^{2} +4xy+8xy+32x^{2}$
$\displaystyle =y( y+4x) +8x( y+4x)$
$\displaystyle =( y+4x)( y+8x)$
$\displaystyle =\left( x^{2} +6+4x\right)\left( x^{2} +6+8x\right)$
$\displaystyle =\left( x^{2} +4x+6\right)\left( x^{2} +8x+6\right)$