Answer
$${x^4} - 3{x^2} + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{4{x^4} - 6{x^2}}}{x}dx} \cr
& {\text{split the numerator}} \cr
& = \int {\left( {\frac{{4{x^4}}}{x} - \frac{{6{x^2}}}{x}} \right)dx} \cr
& = \int {\left( {4{x^3} - 6x} \right)dx} \cr
& {\text{use power rule for indefinite integrals}} \cr
& = 4\left( {\frac{{{x^4}}}{4}} \right) - 6\left( {\frac{{{x^2}}}{2}} \right) + C \cr
& {\text{simplify}} \cr
& = {x^4} - 3{x^2} + C \cr
& {\text{check by differentiation}} \cr
& {\text{ = }}\frac{d}{{dx}}\left( {{x^4} - 3{x^2} + C} \right) \cr
& = 4{x^3} - 6x \cr} $$
