"solving a polynomial equation"
x^3+64=0

so far, i have:
(x)^3+64^(3)
(x+64)(x^2-64x+64^2)
(x+64)(x^2-64x+4096)
x=-64

In the second line you have:
( x + 64 ) ( x² - 64 x + 64² ) 
and it should be:
( x + 4 ) ( x² - 4 x + 4² )
because: ( A³ + B³ )= ( A + B ) ( A² - AB + B² )
You have: B³ = 64 so B = ∛64 = 4
After that:  
x + 4 = 0
x = - 4 
We can justify it:
( - 4 )³ + 61 = 0
- 64 + 64 = 0  ( correct )