Answer
$$48$$
Work Step by Step
$$\eqalign{
& \int_{ - 3}^0 {\left( {{x^2} - 4x + 7} \right)} dx \cr
& {\text{find the antiderivative by the power rule}} \cr
& = \left( {\frac{{{x^{2 + 1}}}}{{2 + 1}} - 4\left( {\frac{{{x^{1 + 1}}}}{{1 + 1}}} \right) + 7x} \right)_{ - 3}^0 \cr
& = \left( {\frac{{{x^3}}}{3} - 2{x^2} + 7x} \right)_{ - 3}^0 \cr
& {\text{part 1 of fundamental theorem of calculus}} \cr
& = \left( {\frac{{{{\left( 0 \right)}^3}}}{3} - 2{{\left( 0 \right)}^2} + 7\left( 0 \right)} \right) - \left( {\frac{{{{\left( { - 3} \right)}^3}}}{3} - 2{{\left( { - 3} \right)}^2} + 7\left( { - 3} \right)} \right) \cr
& {\text{simplify}} \cr
& = \left( 0 \right) - \left( { - 48} \right) \cr
& = 48 \cr} $$
