Answer
$$ - 9$$
Work Step by Step
$$\eqalign{
& \int_{ - 1}^2 {4x\left( {1 - {x^2}} \right)} dx \cr
& {\text{multiply}} \cr
& \int_{ - 1}^2 {\left( {4x - 4{x^3}} \right)} dx \cr
& {\text{find the antiderivative by the power rule}} \cr
& = \left[ {2{x^2} - {x^4}} \right]_{ - 1}^2 \cr
& {\text{part 1 of fundamental theorem of calculus}} \cr
& = \left( {2{{\left( 2 \right)}^2} - {{\left( 2 \right)}^4}} \right) - \left( {2{{\left( { - 1} \right)}^2} - {{\left( { - 1} \right)}^4}} \right) \cr
& {\text{simplify}} \cr
& = \left( {8 - 16} \right) - \left( {2 - 1} \right) \cr
& = - 8 - 1 \cr
& = - 9 \cr} $$
