\(a,x^2-7x+6=x^2-6x-x+6\)
\(=x\left(x-6\right)-\left(x-6\right)=\left(x-1\right)\left(x-6\right)\)
\(b,x^4+64=\left(x^4+16x^2+64\right)-16x^2\)
\(=\left(x^2+8\right)^2-\left(4x\right)^2=\left(x^2-4x+8\right)\left(x^2+4x+8\right)\)
\(c,x^2+12x+35=x^2+5x+7x+35\)
\(=x\left(x+5\right)+7\left(x+5\right)=\left(x+7\right)\left(x+5\right)\)
\(d,4x^4+1=\left(4x^4+4x^2+1\right)-4x^2\)
\(=\left(2x^2+1\right)-\left(2x\right)^2=\left(2x^2+2x+1\right)\left(2x^2-2x+1\right)\)
\(e,x^2-x-56=x^2-8x+7x-56\)
\(=x\left(x-8\right)+7\left(x-8\right)=\left(x+7\right)\left(x-8\right)\)
\(f,4x^2+81=\left(4x^4+36x^2+81\right)-36x^2\)
\(=\left(2x^2+9\right)^2-\left(6x\right)^2=\left(2x^2+6x+9\right)\left(2x^2-6x+9\right)\)
\(g,5x^2-x-4=5x^2-5x+4x-4\)
\(=5x\left(x-1\right)+4\left(x-1\right)=\left(5x+4\right)\left(x-1\right)\)
\(h,64x^4+y^4=\left(64x^4+16x^2y^2+y^4\right)-16x^2y^2\)
\(=\left(8x^2+y^2\right)^2-\left(4xy\right)^2=\left(8x^2+y^2+4xy\right)\left(8x^2+y^2-4xy\right)\)
a) \(x^2-7x+6=x^2-6x-x+6\)
\(=x\left(x-6\right)-\left(x-6\right)\)
\(=\left(x-6\right)\left(x-1\right)\)
b) \(x^4+64=x^4+64+16x^2-16x^2\)
\(=\left(x^2+8\right)^2-\left(4x\right)^2\)
\(=\left(x^2+8-4x\right)\left(x^2+8+4x\right)\)
c) \(x^2+12x+35=x^2+5x+7x+35\)
\(=x\left(x+5\right)+7\left(x+5\right)\)
\(=\left(x+5\right)\left(x+7\right)\)
d) \(4x^4+1=4x^4+1+4x^2-4x^2\)
\(=\left(2x^2+1\right)^2-\left(2x\right)^2\)
\(=\left(2x^2+1-2x\right)\left(2x^2+1+2x\right)\)
e) \(x^2-x-56=x^2+7x-8x-56\)
\(=x\left(x+7\right)-8\left(x+7\right)\)
\(=\left(x+7\right)\left(x-8\right)\)
f) \(4x^4+81=4x^4+81+36x^2-36x^2\)
\(=\left(2x^2+9\right)^2-\left(6x\right)^2\)
\(=\left(2x^2+9-6x\right)\left(2x^2+9+6x\right)\)
g) \(5x^2-x-4=5x^2+4x-5x-4\)
\(=x\left(5x+4\right)-\left(5x+4\right)\)
\(=\left(5x+4\right)\left(x-1\right)\)
h) \(64x^4+y^4=64x^4+y^4+16x^2y^2-16x^2y^2\)
\(=\left(8x^2+y^2\right)^2-\left(4xy\right)^2\)
\(=\left(8x^2+y^2-4xy\right)\left(8x^2+y^2+4xy\right)\)
