\(25-x^2+4xy-4y^2=5^2-\left(x-2y\right)^2=\left(5-x+2y\right)\left(5+x-2y\right)\)
\(x^4-4x^3+4x^2=x^2\left(x^2-4x+4\right)=x^2\left(x-2\right)^2\)
\(x^3-x^2-x+1=x^2\left(x-1\right)-\left(x-1\right)=\left(x^2-1\right)\left(x-1\right)=\left(x+1\right)\left(x-1\right)\left(x-1\right)=\left(x+1\right)\left(x-1\right)^2\)
\(a^5+27a^2=a^2\left(a^3+27\right)=a^2\left(a+3\right)\left(a^2-3a+9\right)\)
\(x^3+3x^2-3x-1=\left(x-1\right)\left(x^2+x+1\right)+3x\left(x-1\right)=\left(x-1\right)\left(x^2+x+1+3x\right)=\left(x-1\right)\left(x^2+4x+1\right)\)
\(4a^2b^2-\left(a^2+b^2-1\right)^2=\left(2ab+a^2+b^2-1\right)\left(2ab-a^2-b^2+1\right)=\left[\left(a+b\right)^2-1\right]\left[1-\left(a-b\right)^2\right]\)
\(\left(a+b-1\right)\left(a+b+1\right)\left(1+a-b\right)\left(1-a+b\right)\)
e)25-x2+4xy-4y2
=25-(x2-4xy+4y2)
=52-(x-y)2
=(5+x-y)(5-x+y)
e) \(25-x^2+4xy-4y^2=5^2-\left[x^2+4xy-\left(2y\right)^2\right]=5^2-\left(x-2y\right)^2=\left(5-x+2y\right)\left(5+x-2y\right)\)
f) \(x^4-4x^3+4x^2=x^2\left(x^2-4x+4\right)=x^2\left(x^2-4x+2^2\right)=x^2\left(x-2\right)^2\)
g) \(x^3-x^2-x+1=\left(x^3-x^2\right)-\left(x-1\right)=x^2\left(x-1\right)-\left(x-1\right)=\left(x-1\right)\left(x^2-1\right)\)
h) \(a^5+27a^2=a^2\left(a^3+27\right)=a^2\left(a^3+3^3\right)=a^2\left(a+3\right)\left(a^2-3a+9\right)\)
i) \(x^3+3x^2-3x-1=\left(x^3-1\right)+\left(3x^2-3x\right)=\left(x-1\right)\left(x^2+x+1\right)+3x\left(x-1\right)=\left(x-1\right)\left(x^2+x+1+3x\right)=\left(x-1\right)\left(x^2+4x+1\right)\)k) \(4a^2b^2-\left(a^2+b^2-1\right)^2=\left(2ab\right)^2-\left(a^2+b^2-1\right)^2=\left(2ab-a^2-b^2+1\right)\left(2ab+a^2+b^2-1\right)=\left[-\left(a+b\right)^2+1\right]\left[\left(a+b\right)^2-1\right]\)
\(=\left[1-\left(a+b\right)^2\right]\left[\left(a+b\right)^2-1\right]=\left(1-a-b\right)\left(1+a+b\right)\left(a+b-1\right)\left(a+b+1\right)=\left(a+b+1\right)^2\left(1-a-b\right)\left(a+b-1\right)\)
