Answer
$124+111+100+91$
$=426$
Work Step by Step
$\sum_{i=1}^4\frac{{x_i}^3+1000}{x_i+10}$
$=\frac{{x_1}^3+1000}{x_1+10}+\frac{{x_2}^3+1000}{x_2+10}+\frac{{x_3}^3+1000}{x_3+10}+\frac{{x_4}^3+1000}{x_4+10}$
$=\frac{(-2)^3+1000}{-2+10}+\frac{(-1)^3+1000}{-1+10}+\frac{0^3+1000}{0+10}+\frac{1^3+1000}{1+10}$
$=\frac{992}{8}+\frac{999}{9}+\frac{1000}{10}+\frac{1001}{11}$
$=124+111+100+91$
$=\boxed{426}$