The solution #1.# Use the power formula #int u^n*du=u^(n+1)/(n+1)#
#int_(-7)^0 (x+1)^2*dx#
Let #u=x+1# and #n=2# and #du=d(x+1)=dx#
#int_(-7)^0 (x+1)^2dx=[(x+1)^(2+1)/(2+1)]_(-7)^0#
#int_(-7)^0 (x+1)^2dx=[((x+1)^3)/3]_(-7)^0#
#int_(-7)^0 (x+1)^2dx=((0+1)^3)/3-((-7+1)^3)/3#
#int_(-7)^0 (x+1)^2dx=1/3-(-216)/3#
#color(red)(int_(-7)^0 (x+1)^2dx=217/3)#
The solution #2.# Expand the integrand
#int_(-7)^0 (x+1)^2dx=int_(-7)^0 (x^2+2x+1)*dx#
#int_(-7)^0 (x+1)^2dx=int_(-7)^0 x^2*dx+int_(-7)^0 2x dx+int_(-7)^0*dx#
#int_(-7)^0 (x+1)^2dx=[x^3/3+ x^2 +x]_(-7)^0#
#int_(-7)^0 (x+1)^2dx=[x^3/3+ x^2 +x]_(-7)^0#
#int_(-7)^0 (x+1)^2dx=[0^3/3+ 0^2 +0-((-7)^3/3+ (-7)^2 +(-7))]#
#int_(-7)^0 (x+1)^2dx=[0-(-343/3+ 49 -7)]#
#int_(-7)^0 (x+1)^2dx=+343/3- 49 +7#
#color(red)(int_(-7)^0 (x+1)^2dx=217/3)#
God bless....I hope the explanation is useful.