How do you find the derivative of f(x)=(x^5+6x^2-1)(1-3x)^2?

I would use the Product and Chain Rule (to deal with the ${\left(\right)}^{2}$):
$f ' \left(x\right) = \left(5 {x}^{4} + 12 x\right) {\left(1 - 3 x\right)}^{2} + \left({x}^{5} + 6 {x}^{2} - 1\right) 2 \left(1 - 3 x\right) \left(- 3\right) =$
$= \left(1 - 3 x\right) \left[\left(5 {x}^{4} + 12 x\right) \left(1 - 3 x\right) - 6 \left({x}^{5} + 6 {x}^{2} - 1\right)\right] =$
$= \left(1 - 3 x\right) \left[- 21 {x}^{5} + 5 {x}^{4} - 72 {x}^{2} + 12 x + 6\right]$