We're given
[tex]\displaystyle \int_4^{-10} g(x) \, dx = -3[/tex]
which immediately tells us that
[tex]\displaystyle \int_{-10}^4 g(x) \, dx = 3[/tex]
In other words, swapping the limits of the integral negates its value.
Also,
[tex]\displaystyle \int_4^6 g(x) \, dx = 5[/tex]
The integral we want to compute is
[tex]\displaystyle \int_{-10}^6 g(x) \, dx[/tex]
which we can do by splitting up the integral at x = 4 and using the known values above. Then the integral we want is
[tex]\displaystyle \int_{-10}^6 g(x) \, dx = \int_{-10}^4 g(x) \, dx + \int_4^6 g(x) \, dx = 3 + 5 = \boxed{8}[/tex]
