a)\(x^3y^3+x^2y^2+4\)
\(=x^3y^3-x^2y^2+2xy+2x^2y^2-2xy+4\)
\(=xy\left(x^2y^2-xy+2\right)+2\left(x^2y^2-xy+2\right)\)
\(=\left(xy+2\right)\left(x^2y^2-xy+2\right)\)
b)\(x^4+x^3+6x^2+5x+5\)
\(=x^4+x^2+x^2+5x^2+5x+5\)
\(=x^2\left(x^2+x+1\right)+5\left(x^2+x+1\right)\)
\(=\left(x^2+5\right)\left(x^2+x+1\right)\)
c)\(x^4-2x^3-12x^2+12x+36\)
\(=x^4-2x^3-6x^2-6x^2+12x+36\)
\(=x^2\left(x^2-2x-6\right)-6\left(x^2-2x-6\right)\)
\(=\left(x^2-6\right)\left(x^2-2x-6\right)\)
d)\(x^8y^8+x^4y^4+1\)
\(=x^8y^8+2x^4y^4+1-x^4y^4\)
\(=\left(x^4y^4+1\right)^2-\left(x^2y^2\right)^2\)
\(=\left(x^4y^4+1+x^2y^2\right)\left(x^4y^4+1-x^2y^2\right)\)
\(=\left(x^4y^4+2x^2y^2+1-x^2y^2\right)\left(x^4y^4+1-x^2y^2\right)\)
\(=\left(\left(x^2y^2+1\right)^2-\left(xy\right)^2\right)\left(x^4y^4+1-x^2y^2\right)\)
\(=\left(x^2y^2+1-xy\right)\left(x^2y^2+1+xy\right)\left(x^4y^4+1-x^2y^2\right)\)
