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\(a^2+b^2+2a-2b-2ab=a^2-2ab+b^2+2\left(a-b\right)\)
\(=\left(a-b\right)^2+2\left(a-b\right)\)
\(=\left(a-b\right)\left(a-b+2\right)\)
\(4a^2-4b^2-4a+1=4a^2-4a+1-\left(2b\right)^2\)
\(=\left(2a-1\right)^2-\left(2b\right)^2\)
\(=\left(2a-1-2b\right)\left(2a-1+2b\right)\)
1)
b) \(\left(x-z\right)^2-y^2+2y-1\)
\(=\left(x^2-2xz+z^2\right)-\left(y-1\right)^2\)
\(=\left(y-z\right)^2-\left(y-1\right)^2\)
\(=\left[\left(x-z\right)+\left(y-1\right)\right]\cdot\left[\left(x-z\right)-\left(y+1\right)\right]\)
\(=\left(x-z+y-1\right)\cdot\left(x-z-y-1\right)\)
a)=(a2+2ab+b2) +(b2-c2) +(ab+ac)-c2
=(a+b)2 -c2 +(b+c)(b-c) +a(b+c)
=(a+b-c)(a+b+c)+(b+c)(a+b-c)
=(a+b-c)(a+2b+2c)
c)a4+2a3+1
=a4 +a3+a3+a2-a2-a+a+1
=a3(a+1)+a2(a+1)-a(a+1)+(a+1)
=(a+1)(a3+a2-a+1)
d)x5+x+1
=(x5+x4+x3)-x4-x3-x2+x2+x+1
=x3(x2+x+1) -x2(x2+x+1) +(x2+x+1)
=(x2+x+1)((x3-x2+1)
e)x8+x4+1
=(x4)2 +2x4+1-x4
=(x4+1)2 -x4
=(x4+1+x2)(x4+1-x2)
=(x4+2x2+1-x2)(x4-x2+1)
=[(x2+1)2-x2 ](x4-x2+1)
=(x2+1-x)(x2+1 )(x4-x2+1)
2a2b2+2a2c2+2b2c2-a4-b4-c4
=4a2b2-(a4+2a2b2+b4)+(2b2c2+2a2c2)-c4
=2(ab)2-(a+b)2+2c2(a2+b2)+c4
=2(ab)2-[(a+b)2-2c2(a2+b2)+c4]
=2(ab)2-(b2+a2-c2)2
=[(a+b)2-c2][-(a-b)2+c2]
=(a+b-c)(a+b+c)(c-a+b)(a+c-b)
\(2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4\)
\(=4a^2b^2-\left(a^4+2a^2b^2+b^4\right)+\left(2b^2c^2+2a^2c^2\right)-c^4\)
\(=2\left(ab\right)^2-\left(a+b\right)^2+2c^2\left(a^2+b^2\right)+c^4\)
\(=2\left(ab\right)^2-\left[\left(a+b\right)^2-2c^2\left(a^2+b^2\right)+c^4\right]\\ =2\left(ab\right)^2-\left(b^2+a^2-c^2\right)^2\)
=\(\left[\left(a+b\right)^2-c^2\right]\left[-\left(a-b\right)^2+c^2\right]\\ =\left(a+b+c\right)\left(a+b+c\right)\left(c-a+b\right)\left(a+c-b\right)\)
a)\(\left(a^3-b^3\right)+\left(a-b\right)^2\)
\(=\left(a-b\right)\left(a^2+ab+b^2\right)+\left(a-b\right)^2\)
\(\left(a-b\right)\left(a^2+ab+b^2+a-b\right)\)
b) \(\left(8a^3-27b^3\right)-2a\left(4a^2-9b^2\right)\)
\(=\left(2a-3b\right)\left(4a^2+6ab+9b^2\right)-2a\left(2a-3b\right)\left(2a+3b\right)\)
\(=\left(2a-3b\right)\left(4a^2+6ab+9b^2-4a^2-6ab\right)\)
\(=\left(2a-3b\right)\cdot9b^2\)
\(=\left(a-b\right)\left(a^2+ab+b^2\right)+a^2-2ab+b^2\)
= ...........
a) 2 - 4a + 4bc + 2a2 - 2b2 - 2c2
=> 2.( 1 - 2a + 2bc + a2 - b2 - c2)
b) 3x2 + 9cd.(2 - 3cd) - 30xy - 3 + 75y2
=> 3x2 + 18cd - 27c2d2 - 30xy - 3 + 75y2
=> 3.( x2 + 6cd - 9c2d2 - 10xy -1 + 25y2)