a)$x^{2}$ - 2xy +$y^{2}$ - $m^{2}$ + 2mn - $n^{2}$
= ($x^{2}$ - 2xy +$y^{2}$) - ( $m^{2}$ - 2mn + $n^{2}$ )
= $(x-y)^{2}$ - $(m-n)^{2}$
= ( x - y + m - n ) ( x - y - m + n )
b) $a^{2}$ - 10a + 25 - $y^{2}$ -4yz - $4z^{2}$
= ( $a^{2}$ - 10a + 25 ) - ( $y^{2}$ + 4yz + $4z^{2}$ )
= ( $a^{2}$ - 10a + $5^{2}$ ) - [$y^{2}$ + 4yz + $(2z)^{2}$]
= $(a-5)^{2}$ - $(y+2z)^{2}$
= ( a - 5 + y + 2z ) ( a - 5 - y - 2z )
c) $x^{2}$ - 2x - 3
= $x^{2}$ + x - 3x - 3
= ( $x^{2}$ + x ) - ( 3x + 3 )
= x ( x + 1 ) - 3 ( x + 1 )
= ( x + 1 ) ( x - 3 )
d)$a^{4}$ + $5a^{3}$ + 15a - 9
= $a^{4}$ + $5a^{3}$ - $3a^{2}$ + $3a^{2}$ + 15a - 9
= ( $a^{4}$ + $5a^{3}$ - $3a^{2}$ ) + ( $3a^{2}$ + 15a - 9 )
= $a^{2}$ ( $a^{2}$ + 5a - 3 ) + 3 ( $a^{2}$ + 5a - 3 )
= ( $a^{2}$ + 5a - 3 ) ( $a^{2}$ + 3 )
e) bc ( b + c ) + ac ( c - a ) - ab ( a + b )
= = $b^{2}$c + $bc^{2}$ + $ac^{2}$ - $a^{2}$c - $a^{2}$b - $ab^{2}$
g) 4x - 4y + $x^{2}$ - 2xy + $y^{2}$
=( 4x - 4y ) + ( $x^{2}$ - 2xy + $y^{2}$ )
= 4 ( x - y ) + $(x-y)^{2}$
= ( x- y ) ( x - y + 4 )
