To solve the equation (x^2-5x)(x+3)(x-8)+108=0, we first expand the expression in the parentheses:

(x^2-5x)(x+3)(x-8) = x^2(x+3)(x-8) - 5x(x+3)(x-8)

Expanding further:

x^2(x+3)(x-8) = x^2(x^2 - 8x + 3x - 24) = x^2(x^2 - 5x - 24) = x^4 - 5x^3 - 24x^2

-5x(x+3)(x-8) = -5x(x^2 - 8x + 3x - 24) = -5x(x^2 - 5x - 24) = -5x^3 + 25x^2 + 120x

Now we can substitute these expanded expressions back into the original equation:

(x^4 - 5x^3 - 24x^2) + (-5x^3 + 25x^2 + 120x) + 108 = 0

Combine like terms:

x^4 - 10x^3 + x^2 + 120x + 108 = 0

This is a quartic equation that cannot be easily solved by factoring. You may need to use numerical methods or a software tool to find the root(s) of this equation.