\(\left(x+3\right)^4-\left(x-3\right)^4-24x^3=108\)

\(\Leftrightarrow\left(x^2+6x+9\right)^2-\left(x^2-6x+9\right)^2-24x^3=108\)

\(\Leftrightarrow12x\left(2x^2+18\right)-24x^3=108\)

\(\Leftrightarrow24x^3+216x-24x^3=108\Leftrightarrow216x=108\Leftrightarrow x=\dfrac{1}{2}\)

\(\left(x+3\right)^4-\left(x-3\right)^4-24x^3=108\\ \Rightarrow\left(\left(x+3\right)^2-\left(x-3\right)^2\right)\left(\left(x+3\right)^2+\left(x-3\right)^2\right)-24x^3=108\\ \Rightarrow12x.\left(\left(x^2+6x+9\right)-\left(x^2-6x+9\right)\right)-24x^3=108\\ \Rightarrow12x.12x-24x^3=108\\ \Rightarrow144x^2-24x^3=108\\ \Rightarrow24x^2\left(6-x\right)=108\\ \)