a) \(100x^2-\left(x^2+25\right)^2\)
\(=\left(10x\right)^2-\left(x^2+25\right)^2\)
\(=\left(10x+x^2+25\right)\left(10x-x^2-25\right)\)
\(=\left(x+5\right)^2\left(10x-x^2-25\right)\)
b) \(1+\left(x-y+5\right)^2-2\left(x-y+5\right)\)
\(=\left[\left(x-y+5\right)-1\right]^2\)
\(=\left(x-y+4\right)^2\)
c) \(\left(x^2+4y^2-5\right)^2-16\left(x^2y^2+2xy+1\right)\)
\(=\left(x^2+4y^2-5\right)^2-4^2\left(xy+1\right)^2\)
\(=\left[\left(x^2+4y^2-5\right)-4\left(xy+1\right)\right]\left[\left(x^2+4y^2-5\right)+4\left(xy+1\right)\right]\)
\(=\left(x^2+4y^2-5-4xy-4\right)\left(x^2+4y^2-5+4xy+4\right)\)
\(=\left(x^2-4xy+4y^2-9\right)\left(x^2+4xy+4y^2-1\right)\)
\(=\left[\left(x-2y\right)^2-3^2\right]\left[\left(x+2y\right)^2-1^2\right]\)
\(=\left(x-2y-3\right)\left(x-2y+3\right)\left(x+2y-1\right)\left(x+2y+1\right)\)
d) \(\left(x^2+8x-34\right)^2-\left(3x^2-8x-2\right)^2\)
\(=\left[\left(x^2+8x-34\right)-\left(3x^2-8x-2\right)\right]\left[\left(x^2+8x-34\right)+\left(3x^2-8x-2\right)\right]\)
\(=\left(x^2+8x-34-3x^2+8x+2\right)\left(x^2+8x-34+3x^2-8x-2\right)\)
\(=\left(-2x^2+16x-32\right)\left(4x^2-36\right)\)
\(=-2\left(x^2-8x+16\right)\left[\left(2x\right)^2-6^2\right]\)
\(=-2\left(x-4\right)^2\left(2x-6\right)\left(2x+6\right)\)
\(=-2\left(x-4\right)^24\left(x-3\right)\left(x+3\right)\)
\(=8\left(x-4\right)^2\left(x-3\right)\left(x+3\right)\)
