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a) (a+b+c)^2 + (a+b-c)^2 - 4c^2
\(=\left(a+b+c\right)^2+\left[\left(a+b-c\right)^2-\left(2c\right)^2\right]\)
\(=\left(a+b+c\right)^2+\left(a+b-c+2c\right)\left(a+b-c-2c\right)\)
\(=\left(a+b+c\right)^2+\left(a+b+c\right)\left(a+b-3c\right)\)
\(=\left(a+b+c\right)\left(a+b+c+a+b-3c\right)\)
\(=\left(a+b+c\right)\left(2a+2b-2c\right)\)
\(=2\left(a+b+c\right)\left(a+b-c\right)\)
b) 4a^2b^2 - (a^2+b^2-c^2)^2
\(=\left(2ab\right)^2-\left(a^2+b^2-c^2\right)^2=\left(2ab+a^2+b^2-c^2\right)\left(2ab-a^2-b^2+c^2\right)\)
\(=\left[\left(a^2+2ab+b^2\right)-c^2\right]\left[c^2-\left(a^2-2ab+b^2\right)\right]\)
\(=\left[\left(a+b\right)^2-c^2\right]\left[c^2-\left(a-b\right)^2\right]\)
\(=\left(a+b+c\right)\left(a+b-c\right)\left(c+a-b\right)\left(c-a+b\right)\)
c) a(b^3-c^3) + b(c^3-a^3) + c(a^3-b^3)
\(=ab^3-ac^3+bc^3-a^3b+a^3c-b^3c\)
\(=a^3\left(c-b\right)+bc\left(c-b\right)\left(c+b\right)-a\left(c-b\right)\left(c^2+bc+b^2\right)\)
\(=a^3\left(c-b\right)+\left(c-b\right)\left(bc^2+b^2c\right)-\left(c-b\right)\left(ac^2+abc+ab^2\right)\)
\(=\left(c-b\right)\left(a^3+bc^2+b^2c-ac^2-abc-ab^2\right)\)
a) (a+b+c)^2 + (a+b-c)^2 - 4c^2
\(=\left(a+b+c\right)^2+\left[\left(a+b-c\right)^2-\left(2c\right)^2\right]\)
\(=\left(a+b+c\right)^2+\left(a+b-c+2c\right)\left(a+b-c-2c\right)\)
\(=\left(a+b+c\right)^2+\left(a+b+c\right)\left(a+b-3c\right)\)
\(=\left(a+b+c\right)\left(a+b+c+a+b-3c\right)\)
\(=\left(a+b+c\right)\left(2a+2b-2c\right)\)
\(=2\left(a+b+c\right)\left(a+b-c\right)\)
b) 4a^2b^2 - (a^2+b^2-c^2)^2
\(=\left(2ab\right)^2-\left(a^2+b^2-c^2\right)^2=\left(2ab+a^2+b^2-c^2\right)\left(2ab-a^2-b^2+c^2\right)\)
\(=\left[\left(a^2+2ab+b^2\right)-c^2\right]\left[c^2-\left(a^2-2ab+b^2\right)\right]\)
\(=\left[\left(a+b\right)^2-c^2\right]\left[c^2-\left(a-b\right)^2\right]\)
\(=\left(a+b+c\right)\left(a+b-c\right)\left(c+a-b\right)\left(c-a+b\right)\)
c) a(b^3-c^3) + b(c^3-a^3) + c(a^3-b^3)
\(=ab^3-ac^3+bc^3-a^3b+a^3c-b^3c\)
\(=a^3\left(c-b\right)+bc\left(c-b\right)\left(c+b\right)-a\left(c-b\right)\left(c^2+bc+b^2\right)\)
\(=a^3\left(c-b\right)+\left(c-b\right)\left(bc^2+b^2c\right)-\left(c-b\right)\left(ac^2+abc+ab^2\right)\)
\(=\left(c-b\right)\left(a^3+bc^2+b^2c-ac^2-abc-ab^2\right)\)
\(M=\left(a^2+b^2-c^2\right)^2-4a^2b^2\)
\(M=\left(a^2+b^2-c^2\right)^2-\left(2ab\right)^2\)
\(M=\left(a^2+b^2-c^2-2ab\right)\left(a^2+b^2-c^2+2ab\right)\)
\(M=\left(\left(a^2-2ab+b^2\right)-c^2\right)\left(\left(a^2+2ab+b^2\right)-c^2\right)\)
\(M=\left(\left(a-b\right)^2-c^2\right)\left(\left(a+b\right)^2-c^2\right)\)
\(M=\left(a-b-c\right)\left(a-b+c\right)\left(a+b-c\right)\left(a+b+c\right)\)
4a2b2-(a2+b2-c2)2
= (4ab-a2-b2+c2)(4ab+a2+b2-c2)
= -[(a-b)2-c2][(a+b)2-c2]
=-(a-b+c)(a-b-c)(a+b-c)(a+b+c)
=(b-a-c)(b+c-a)(a+b-c)(a+b+c)
\(4a^2b^2-\left(a^2+b^2-c^2\right)^2\)
\(=\left(2ab\right)^2-\left(a^2+b^2-c^2\right)^2\)
\(=\left(2ab-a^2-b^2+c^2\right)\left(2ab+a^2+b^2-c^2\right)\)
a, 4a^2b^3 - 6a^3b^2 = 2a^2b^2(2b - 3a)
b, 5(a + b) +x( a + b ) = ( 5 + x )( a + b )
c, (a - b)^2 - ( b - a ) = ( a - b )^2 + ( a - b ) = (a - b) ( a - b + 1)
M = ( a2 + b2 - c2 )2 - 4a2b2
= ( a2 + b2 - c2 )2 - ( 2ab )2 = (a2 + b2 - c2 + 2ab )( a2 + b2 - c2 - 2ab )
= [( a + b )2 - c2 ] . [( a - b )2 -c2 ]
= ( a + b + c )( a+ b - c )( a - b + c )( a - b -c )