As it stands, the only real solution is x=-5.34 approx. But I suspect there is an error in the question. I think that one of the signs is wrong. Perhaps it should be X^3+3X^2-8X-24=0?
A solution is found by regrouping the terms:
X^3-8X+3X^2-24=0. Now we introduce some factors:
X(X^2-8)+3(X^2-8)=0. There is now a common factor X^2-8 so we have
(X+3)(X^2-8)=0, so X=-3 is one solution.
Also X^2=8 is another, and if we take square roots we get X=+sqrt(8) or -sqrt(8). sqrt(8) can be written sqrt(4*2)=sqrt(4)*sqrt(2)=2sqrt(2), so X=2sqrt(2) or -2sqrt(2).
The equation could have been all pluses between the terms (X^3+3X^2+8X+24=0) making the answer (X+3)(X^2+8)=0 so X=-3 is the only real solution. The other two solutions would be imaginary: X=2isqrt(2) or -2isqrt(2) where i is the imaginary square root of -1.