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)$x^8y^8+x^4y^4+1$

$=x^8y^8+2x^4y^4+1-x^4y^4$

$=\left(x^4y^4+1\right)^2-\left(x^2y^2\right)^2$

$=\left(x^4y^4+1+x^2y^2\right)\left(x^4y^4+1-x^2y^2\right)$

$=\left(x^4y^4+2x^2y^2+1-x^2y^2\right)\left(x^4y^4+1-x^2y^2\right)$

$=\left(\left(x^2y^2+1\right)^2-\left(xy\right)^2\right)\left(x^4y^4+1-x^2y^2\right)$

$=\left(x^2y^2+1-xy\right)\left(x^2y^2+1+xy\right)\left(x^4y^4+1-x^2y^2\right)$