Hello turbo,
(I) $\displaystyle 2x^2-2y^2$
Always look for common monimial factors over the terms of the polynomial. In this case, 2 is a common factor. Now, we have
$\displaystyle 2(x^2-y^2)$
You should recognize the difference of two squares. You should memorize the pattern for factoring it. I'll leave the finish for you.
(II) $\displaystyle 64x^2+16xy+y^2$
You may or may not recognize this as a perfect square trinomial, but you should see that the first and last terms are perfect squares. You should also memorize a factorization pattern for this.
$\displaystyle x^2+2xy+y^2=(x+y)^2$
You can do the rest.
(III) $\displaystyle 36x^2z-21xyz+15yxz-6y^2z$
You should notice that the two middle terms are like terms and can be combined (even though the variables are in a different order).
$\displaystyle 36x^2z-6xyz-6y^2z$
Now, find your monomial factor of 6z and factor it out.
$\displaystyle 6z(6x^2-xy-y^2)$
See if you can finish factoring the trinomial.