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\(a.\left(b^2+c^2+bc\right)+b.\left(c^2+a^2+ac\right)+c.\left(a^2+b^2+ab\right)\)
\(=ab^2+ac^2+abc+bc^2+ba^2+bac+ca^2+cb^2+cab\)
\(=\left(ab^2+ba^2+abc\right)+\left(ac^2+ca^2+bac\right)+\left(bc^2+cb^2+cab\right)\)
\(=ab.\left(b+a+c\right)+ac.\left(c+a+b\right)+bc.\left(c+b+a\right)\)
\(=\left(a+b+c\right).\left(ab+ac+bc\right)\)
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a,\(5xy^3-2xyz-15y^2+6z\)
\(=\left(5xy^3-15y^2+6z-2xyz\right)\)
\(=5y^2\left(xy-3\right)-2z\left(xy-3\right)\)
\(=\left(5y^2-2z\right)\left(xy-3\right)\)
a) 5xy3 - 2xyz - 15y2 + 6z
= ( 5xy3 - 15y2 ) - ( 2xyz - 6z )
= 5y2( xy - 3 ) - 2z( xy - 3 )
= ( xy - 3 )( 5y2 - 2z )
b) ab3c2 - a2b2c2 + ab2c3 - a2bc3
= abc2( b2 - ab + bc - ac )
= abc2[ ( b2 - ab ) + ( bc - ac ) ]
= abc2[ b( b - a ) + c( b - a ) ]
= abc2( b - a )( b + c )
\(x^2-y^2+4x+4\)
\(=\left(x+2\right)^2-y^2\)
\(=\left(x+2+y\right)\left(x+2-y\right)\)
\(4x^2-y^2+8\left(y-2\right)\)
\(=4x^2-\left(y^2-8y+16\right)\)
\(=4x^2-\left(y-4\right)^2\)
\(=\left(2x+y-4\right)\left(2x-y+4\right)\)
\(\left(a+b\right)\left(a^2-b^2\right)+\left(b+c\right)\left(b^2-c^2\right)+\left(c+a\right)\left(c^2-a^2\right)\)
\(=\left(a+b\right)\left(a^2-b^2\right)-\left(b+c\right)\left(a^2-b^2\right)-\left(b+c\right)\left(c^2-a^2\right)+\left(a+c\right)\left(c^2-a^2\right)\)
\(=\left(a^2-b^2\right)\left(a+b-b-c\right)-\left(c^2-a^2\right)\left(b+c-c-a\right)\)
\(=\left(a-b\right)\left(a+b\right)\left(a-c\right)-\left(c-a\right)\left(c+a\right)\left(b-a\right)\)
\(=\left(a-b\right)\left(a-c\right)\left(a+b-c-a\right)\)
\(=\left(a-b\right)\left(a-c\right)\left(b-c\right)\)
(a2+b2+ab)2-a2b2-b2c2-c2a2
=a4+b4+a2b2+2.(a2b2+a3b+ab3)-a2b2-b2c2-c2a2
=a4+b4+a2b2+2a2b2+2a3b+2ab3-a2b2-b2c2-c2a2
=a4+b4+2a2b2+2a3b+2ab3-b2c2-c2a2
= (a4+b4+2a2b2)+(2a3b+2ab3)-(c2a2+b2c2)
= (a2+b2)2+2ab(a2+b2)-c2(a2+b2)
= (a2+b2)(a2+b2+2ab-c2)
=(a2+b2)[(a+b)2-c2]
=(a2+b2)(a+b+c)(a+b-c)