$$ \begin{aligned} (x+1)(x-2)(x-8)(x+5)+360 &= 0&& \text{simplify left side} \\[1 em](x^2-2x+x-2)(x-8)(x+5)+360 &= 0&& \\[1 em](x^2-x-2)(x-8)(x+5)+360 &= 0&& \\[1 em](x^3-8x^2-x^2+8x-2x+16)(x+5)+360 &= 0&& \\[1 em](x^3-9x^2+6x+16)(x+5)+360 &= 0&& \\[1 em]x^4-4x^3-39x^2+46x+80+360 &= 0&& \\[1 em]x^4-4x^3-39x^2+46x+440 &= 0&& \\[1 em] \end{aligned} $$

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