SOLUTION: This problem is from my daughter's (Arundhati's) VIII standard text book on advanced mathematics. I checked it through a calculator, the expressions comes to 4, but how to solve it

This problem is from my daughter's (Arundhati's) VIII 
standard text book on advanced mathematics. I checked 
it through a calculator, the expressions comes to 4, 
but how to solve it: 

Prove that 
 

-----------------------------

It is well known that a solution to the cubic equation

 is given by the formula



Since your expression is of this form, let's find the 
values of p and q that would make this equivalent to 
your expression. To do this we equate like parts

 



The first gives q = -40

Substituting that in the second equation:





Squaring both sides:







Multiply both sides by 27



Taking cube roots of both sides:



So your expression is a solution to the cubic
equation   or



By DesCartes rule of signs, this cubic has
just one positive solution, so your expression
must be it.

Direct substitution of 4 into this equation shows
that 4 is its only positive solution:

  

   
          

Since the cubic equation  can have but 
one positive solution, and both your expression and 4 
must be this positive solution, your expression must
equal 4.

Edwin