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 Giải thích các bước giải:

$\displaystyle \begin{array}{{>{\displaystyle}l}} Bài\ 1:\\ a.\ x^{2} +4x+3=x^{2} +3x+x+3\\ =x( x+3) +( x+3) =( x+1)( x+3)\\ b.\ 16x-5x^{2} -3=-5x^{2} +15x+x-3\\ =-5x( x-3) +( x-3) =( -5x+1)( x-3)\\ c.\ 2x^{2} +7x+5=2x^{2} +2x+5x+5\\ =2x( x+1) +5( x+1) =( 2x+5)( x+1)\\ d.\ 2x^{2} +3x-5=2x^{2} -2x+5x-5\\ =2x( x-1) +5( x-1) =( 2x+5)( x-1)\\ e.\ x^{3} -3x^{2} +1-3x=x^{3} +1-3x( x+1)\\ =( x+1)\left( x^{2} -x+1\right) -3x( x+1)\\ =( x+1)\left( x^{2} -x+1-3x\right)\\ =( x+1)\left( x^{2} -4x+1\right)\\ f.\ x^{2} -4x-5=x^{2} -5x+x-5\\ =x( x-5) +( x-5) =( x+1)( x-5)\\ g.\ \left( a^{2} +1\right)^{2} -4a^{2} =\left( a^{2} +1-2a^{2}\right)\left( a^{2} +1+2a^{2}\right)\\ =\left( 1-a^{2}\right)\left( 1+3a^{2}\right)\\ h.\ x^{3} -3x^{2} -4x+12=x^{3} +2x^{2} -5x^{2} -10x+6x+12\\ =x^{2}( x+2) -5x( x+2) +6( x+2)\\ =\left( x^{2} -5x+6\right)( x+2) =\left( x^{2} -3x-2x+6\right)( x+2)\\ =( x( x-3) -2( x-3))( x+2) =( x-2)( x-3)( x+2)\\ i.\ x^{4} +x^{3} +x+1=x^{4} +x+x^{3} +1\\ =x\left( x^{3} +1\right) +\left( x^{3} +1\right) =( x+1)\left( x^{3} +1\right)\\ =( x+1)^{2}\left( x^{2} -x+1\right)\\ k.\ x^{4} -x^{3} -x^{2} +1=x^{3}( x-1) -\left( x^{2} -1\right)\\ =x^{3}( x-1) -( x-1)( x+1) =( x-1)\left( x^{3} -x-1\right)\\ l.\ ( 2x+1)^{2} -( x-1)^{2} =( 2x+1+x-1)( 2x+1-x+1)\\ =3x.( x+2)\\ m.\ x^{4} +4x^{2} -5=x^{4} -x^{2} +5x^{2} -5\\ =x^{2}\left( x^{2} -1\right) +5\left( x^{2} -1\right) =\left( x^{2} +5\right)( x-1)( x+1)\\ Bài\ 2\\ a.\ -x-y^{2} +x^{2} -y=-( x+y) +( x-y)( x+y)\\ =( x-y-1)( x+y)\\ b.\ x( x+y) -5x-5y=x( x+y) -5( x+y)\\ =( x-5)( x+y)\\ c.\ x^{2} -5x+5y-y^{2} =( x-y)( x+y) -5( x-y)\\ =( x-y)( x+y-5)\\ d.\ 5x^{3} -5x^{2} y-10x^{2} +10xy=5x^{2}( x-y) -10x( x-y)\\ =5x( x-2)( x-y)\\ e.\ 27x^{3} -8y^{3} =( 3x-2y)\left( 9x^{2} +12x+4y^{2}\right)\\ f.\ x^{2} -y^{2} -x-y=( x-y)( x+y) -( x+y) =( x+y)( x-y-1)\\ g.\ x^{2} -y^{2} -2xy+y^{2} =x^{2} -2xy=x( x-2y)\\ h.\ x^{2} -y^{2} +4-4x=x^{2} -4x+4-y^{2} =( x-2)^{2} -y^{2}\\ =( x-2-y)( x-2+y)\\ i.\ x^{6} -y^{6} =\left( x^{3} -y^{3}\right)\left( x^{3} +y^{3}\right) =( x-y)\left( x^{2} +xy+y^{2}\right)\left( x^{3} +y^{3}\right)\\ k.\ x^{3} +3x^{2} +3x+1-27z^{3} =( x+1)^{3} -( 3z)^{3}\\ =( x+1-3z)\left(( x+1)^{2} +( x+1) 3z+9z^{2}\right)\\ l.\ 4x^{2} +4x-9y^{2} +1=( 2x+1) -( 3y)^{2} =( 2x+1-3y)( 2x+1+3y)\\ m.\ x^{2} -3x+xy-3y=x( x-3) +y( x-3) =( x+y)( x-3)\\ Bài\ 3:\\ a.\ 5x^{2} -10xy+5y^{2} -20z^{2} =5\left(( x-y)^{2} -( 2z)^{2}\right)\\ =5( x-y-2x)( x-y+2z)\\ b.\ x^{2} -z^{2} +y^{2} -2xy=( x-y)^{2} -z^{2} =( x-y-z)( x-y+z)\\ c.\ a^{3} -ay-a^{2} x+xy=a\left( a^{2} -y\right) -x\left( a^{2} -y\right) =( a-x)\left( a^{2} -y\right)\\ d.\ x^{2} -2xy-4z^{2} +y^{2} =( x+y)^{2} -( 2z)^{2} =( x+y-2z)( x+y+2z)\\ e.\ 3x^{2} -6xy+3y^{2} -12z^{2} =3\left(( x-y)^{2} -( 2z)^{2}\right)\\ =3( x-y-2z)( x-y+2z)\\ f.\ x^{2} -6xy-25z^{2} +9y^{2} =( x-3y)^{2} -( 5z)^{2}\\ =( x-3y-5z)( x-3y+5z)\\ g.\ x^{2} -y^{2} +2yz-z^{2} =x^{2} -( y-z)^{2} =( x-y+z)( x+y-z)\\ h.\ x^{2} -2xy+y^{2} -xz+yz=( x-y)^{2} -z( x-y) =( x-y)( x-y-z)\\ i.\ x^{2} -2xy+tx-2ty=x( x-2y) +t( x-2y) =( x+t)( x-2y)\\ k.\ 2xy+3z+6y+xz=2xy+6y+xz+3z\\ =2y( x+3) +z( x+3) =( 2y+z)( x+3)\\ l.\ x^{2} +2xz+2xy+4yz=x( x+2z) +2y( x+2z) =( x+2y)( x+2z)\\ m.\ ( x+y+z)^{3} -x^{3} -y^{3} -z^{3}\\ =x^{3} +y^{3} +z^{3} +3( x+y)( x+z)( y+z) -z^{3} -y^{3} -z^{3}\\ =3( x+y)( x+z)( y+z) \end{array}$