Prove that the equation has at least a solution in the interval (0,1) 4x^3 + 3x^2 - 2x - 1 = 0

We'll use the Rolle's theorem to prove that the given
equation  has a root over the interval (0,1).

Let's see
how:

We'll choose a Rolle function
f:[0,1]->R

f(x)=x^4+x^3-x^2-x

According to
the Rolle's
rule,

 f(1)-f(0)=f'(c)(1-0)

 where
c belongs to (0,1).

If f(x) is a Rolle function, then 
f(1)=f(0).

f'(c)=0.

We'll
differentiate Rolle's function and we'll
get:

f'(x)=4x^3+3x^2-2x-1

If
f'(c)=0 ,then c is a root of f'(x), c belongs to (0,1).
q.e.d.