Answer
$$y = - 2{x^2} + 2{x^3} + C$$
Work Step by Step
$$\eqalign{
& \frac{{dy}}{{dx}} = - 4x + 6{x^2} \cr
& {\text{Separating variables leads to}} \cr
& dy = - 4xdx + 6{x^2}dx \cr
& {\text{To solve this equation}}{\text{, determine the antiderivative of each side}} \cr
& \int {dy} = - \int {4x} dx + \int {6{x^2}} dx \cr
& {\text{integrate using the power rule}} \cr
& y = - 4\left( {\frac{{{x^2}}}{2}} \right) + 6\left( {\frac{{{x^3}}}{3}} \right) + C \cr
& y = - 2{x^2} + 2{x^3} + C \cr
& \cr
& {\text{verifying that the solution satisfies the original differential equation}} \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ { - 2{x^2} + 2{x^3} + C} \right] \cr
& \frac{{dy}}{{dx}} = - 2\left( {2x} \right) + 2\left( {3{x^2}} \right) + 0 \cr
& \frac{{dy}}{{dx}} = - 4x + 6{x^2} \cr} $$
