\(\eqalign{ & a)\,\,4{x^4} + 4{x^3} - {x^2} - x \cr & = \left( {4{x^4} + 4{x^3}} \right) - \left( {{x^2} + x} \right) \cr & = 4{x^3}\left( {x + 1} \right) - x\left( {x + 1} \right) \cr & = x\left( {x + 1} \right)\left( {4{x^2} - 1} \right) \cr & = x\left( {x + 1} \right)\left( {2x - 1} \right)\left( {2x + 1} \right) \cr & b)\,\,{x^6} - {x^4} - 9{x^3} + 9{x^2} \cr & = \left( {{x^6} - {x^4}} \right) - \left( {9{x^3} - 9{x^2}} \right) \cr & = {x^4}\left( {{x^2} - 1} \right) - 9{x^2}\left( {x - 1} \right) \cr & = {x^4}\left( {x - 1} \right)\left( {x + 1} \right) - 9{x^2}\left( {x - 1} \right) \cr & = {x^2}\left( {x - 1} \right)\left[ {{x^2}\left( {x + 1} \right) - 9} \right] \cr & c)\,\,{\left( {xy + 4} \right)^2} - 4{\left( {x + y} \right)^4} \cr & = {\left( {xy + 4} \right)^2} - {\left[ {2{{\left( {x + y} \right)}^2}} \right]^2} \cr & = \left( {xy + 4 - 2{{\left( {x + y} \right)}^2}} \right)\left( {xy + 4 + 2{{\left( {x + y} \right)}^2}} \right) \cr & d)\,\,4{x^4} + 1 \cr & = 4{x^4} + 4{x^2} + 1 - 4{x^2} \cr & = \left( {2{x^2} + 1} \right) - {\left( {2x} \right)^2} \cr & = \left( {2{x^2} + 1 - 2x} \right)\left( {2{x^2} + 1 + 2x} \right) \cr & e)\,\,{x^4} + 342 \cr & = {x^4} + 36{x^2} + 342 - 36{x^2} \cr & = {\left( {{x^2} + 18} \right)^2} - {\left( {6x} \right)^2} \cr & = \left( {{x^2} + 18 - 6x} \right)\left( {{x^2} + 18 + 6x} \right) \cr & f)\,\,{x^3} - {x^2} - x - 2 \cr & = {x^3} - 8 - {x^2} - x + 6 \cr & = \left( {{x^3} - 8} \right) - \left( {{x^2} + x - 6} \right) \cr & = \left( {x - 2} \right)\left( {{x^2} + 2x + 4} \right) - \left( {{x^2} - 2x + 3x - 6} \right) \cr & = \left( {x - 2} \right)\left( {{x^2} + 2x + 4} \right) - \left[ {x\left( {x - 2} \right) + 3\left( {x - 2} \right)} \right] \cr & = \left( {x - 2} \right)\left( {{x^2} + 2x + 4} \right) - \left( {x - 2} \right)\left( {x + 3} \right) \cr & = \left( {x - 2} \right)\left( {{x^2} + 2x + 4 - x - 3} \right) \cr & = \left( {x - 2} \right)\left( {{x^2} + x + 1} \right) \cr & g)\,\,{x^3} - 9{x^2} + 6x + 16 \cr & = {x^3} + {x^2} - 10{x^2} - 10x + 16x + 16 \cr & = {x^2}\left( {x + 1} \right) - 10x\left( {x + 1} \right) + 16\left( {x + 1} \right) \cr & = \left( {x + 1} \right)\left( {{x^2} - 10x + 16} \right) \cr & = \left( {x + 1} \right)\left( {{x^2} - 2x - 8x + 16} \right) \cr & = \left( {x + 1} \right)\left[ {x\left( {x - 2} \right) - 8\left( {x - 2} \right)} \right] \cr & = \left( {x + 1} \right)\left( {x - 2} \right)\left( {x - 8} \right) \cr & h)\,\,6{x^2} - 11x + 3 \cr & = 6{x^2} - 9x - 2x + 3 \cr & = 3x\left( {2x - 3} \right) - \left( {2x - 3} \right) \cr & = \left( {2x - 3} \right)\left( {3x - 1} \right) \cr & i)\,\,2{x^2} - 5xy - 3{y^2} \cr & = 2{x^2} - 6xy + xy - 3{y^2} \cr & = 2x\left( {x - 3y} \right) + y\left( {x - 3y} \right) \cr & = \left( {x - 3y} \right)\left( {2x + y} \right) \cr} \) \(\eqalign{ & k)\,\,\left( {x + 2} \right)\left( {x + 3} \right)\left( {x + 4} \right)\left( {x + 5} \right) - 24 \cr & = \left( {x + 2} \right)\left( {x + 5} \right)\left( {x + 3} \right)\left( {x + 4} \right) - 24 \cr & = \left( {{x^2} + 7x + 10} \right)\left( {{x^2} + 7x + 12} \right) - 24 \cr & = {\left( {{x^2} + 7x} \right)^2} + 22\left( {{x^2} + 7x} \right) + 120 - 24 \cr & = {\left( {{x^2} + 7x} \right)^2} + 22\left( {{x^2} + 7x} \right) + 96 \cr & = {\left( {{x^2} + 7x} \right)^2} + 6\left( {{x^2} + 7x} \right) + 16\left( {{x^2} + 7x} \right) + 96 \cr & = \left( {{x^2} + 7x} \right)\left( {{x^2} + 7x + 6} \right) + 16\left( {{x^2} + 7x + 6} \right) \cr & = \left( {{x^2} + 7x + 6} \right)\left( {{x^2} + 7x + 16} \right) \cr & l)\,\,{x^2} + 2xy + {y^2} - x - y - 12 \cr & = {\left( {x + y} \right)^2} - \left( {x + y} \right) - 12 \cr & = {\left( {x + y} \right)^2} - 4\left( {x + y} \right) + 3\left( {x + y} \right) - 12 \cr & = \left( {x + y} \right)\left( {x + y - 4} \right) + 3\left( {x + y - 4} \right) \cr & = \left( {x + y - 4} \right)\left( {x + y + 3} \right) \cr} \)

a)4x4+4x3−x2−x=(4x4+4x3)−(x2+x)=4x3(x+1)−x(x+1)=x(x+1)(4x2−1)=x(x+1)(2x−1)(2x+1)

b)x6−x4−9x3+9x2=(x6−x4)−(9x3−9x2)=x4(x2−1)−9x2(x−1)=x4(x−1)(x+1)−9x2(x−1)=x2(x−1)[x2(x+1)−9]

c)(xy+4)2−4(x+y)4=(xy+4)2−[2(x+y)2]2=(xy+4−2(x+y)2)(xy+4+2(x+y)2)

d)4x4+1=4x4+4x2+1−4x2=(2x2+1)−(2x)2=(2x2+1−2x)(2x2+1+2x

)e)x4+342=x4+36x2+342−36x2=(x2+18)2−(6x)2=(x2+18−6x)(x2+18+6x)

f)x3−x2−x−2=x3−8−x2−x+6=(x3−8)−(x2+x−6)=(x−2)(x2+2x+4)−(x2−2x+3x−6)=(x−2)(x2+2x+4)−[x(x−2)+3(x−2)]=(x−2)(x2+2x+4)−(x−2)(x+3)=(x−2)(x2+2x+4−x−3)=(x−2)(x2+x+1)

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