$$x(x+1)(x+2)(x+3)\frac{x+4}{120} = 100$$
Answer
$$ \begin{matrix}x_1 = 4.69742 & x_2 = 0.06882+6.0787i & x_3 = 0.06882-6.0787i \\[1 em] x_4 = -7.41753+3.7559i & x_5 = -7.41753-3.7559i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x(x+1)(x+2)(x+3)\frac{x+4}{120} &= 100&& \text{multiply ALL terms by } \color{blue}{ 120 }. \\[1 em]120x(x+1)(x+2)(x+3)\frac{x+4}{120} &= 120\cdot100&& \text{cancel out the denominators} \\[1 em]x^5+10x^4+35x^3+50x^2+24x &= 12000&& \text{move all terms to the left hand side } \\[1 em]x^5+10x^4+35x^3+50x^2+24x-12000 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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