w^3 + 5w^2 - w = 5

w^3 + 5w^2 - w - 5 = 0

To solve this, we use the factor theorem

f(w) = w^3 + 5w^2 - w -5

If f(a) = 0, then (x-a) is a factor of f(x) --- Don't worry if you don't understand this, I didn't either when I did this for the first time a few years back.

Basically, we are substituting for values of x which will gives us 0 at the end.

f(0) = -5   NOT 0
f(1) = 1 + 5 -1 -5 = 0   --- This is what we want!

Hence, (w-1) is a factor of f(x), which is  w^3 + 5w^2 - w -5 = 0


f(w) = w^3 + 5w^2 - w -5

Now, divide f(w) by (w-1). You can do this on paper but I've figured it out as:

f(w) = (w^2 + 6w + 5)(w-1)

Factorise the quadratic

 (w+1)(w+5)(w-1) = 0

Hence, our solutions are 1,-1, -5