You need to note first two things.
1. The function f(x) need to be a continuous function on [-3,-1] (indeed f(x) is a polynomial function, is continuous on every real number x)
2. Take the closed interval [-3,-1]
Now, evaluate f(-3) and f(-1).
[tex]f(-3) = (-3)^3 + (-3)^2 -2(-3) + 5 = -27 + 9 +6 + 5 = -7 < 0\\f(-1) = (-1)^3 + (-1)^2 -2 (-1) + 5 = -1 + 1 +2 + 5 = 7 >0[/tex]
So, we have f(-3) < 0 and f(-1). So, [tex]f(-3) < 0 < f(-1)[/tex], then by the Intermediate Value Theorem, there exist [tex]c \in (-3,-1)[/tex] such that [tex]f(c) = 0[/tex].
