y = x³ + 6x - 5 is negative when x = 0 and positive when x = 1, so at some point must be zero between these two values, thus, we have a solution in that interval.
How we can show that we have a solution between 0 and 1?
This is a continuous function, so to check that we have a solution between x = 0 and x = 1, we only need to evaluate:
y = x³ + 6x - 5
in x = 0 and in x = 1, and see that we get a negative outcome in one case and a positive one in the other:
when x = 0
y = 0³ + 6*0 - 5 = -5
when x = 1
y = 1³ + 6*1 - 5 = 2
So y = x³ + 6x - 5 must be zero at one point between x = 0 and x = 1.