Answer
$$\frac{{11{x^2}}}{2} - {x^3} + 4x + C$$
Work Step by Step
$$\eqalign{
& \int {\left( {3x + 1} \right)\left( {4 - x} \right)} dx \cr
& {\text{multiply the integrand}}{\text{, use foil}} \cr
& = \int {\left( {12x - 3{x^2} + 4 - x} \right)} dx \cr
& = \int {\left( {11x - 3{x^2} + 4} \right)} dx \cr
& {\text{use power rule for indefinite integrals}} \cr
& = \frac{{11{x^{1 + 1}}}}{{1 + 1}} - \frac{{3{x^{2 + 1}}}}{{2 + 1}} + 4x + C \cr
& = \frac{{11{x^2}}}{2} - \frac{{3{x^3}}}{3} + 4x + C \cr
& {\text{simplify}} \cr
& = \frac{{11{x^2}}}{2} - {x^3} + 4x + C \cr
& \cr
& \cr
& {\text{check the antiderivative by differentiation}} \cr
& {\text{ = }}\frac{d}{{dx}}\left( {\frac{{11{x^2}}}{2} - {x^3} + 4x + C} \right) \cr
& = \frac{{11\left( 2 \right)x}}{2} - 3{x^2} + 4\left( 1 \right) + 0 \cr
& = 11x - 3{x^2} + 4 \cr
& {\text{factor the trinomial}} \cr
& = \left( {3x + 1} \right)\left( {4 - x} \right) \cr} $$
