$$\frac{x+1}{x+8} = \frac{x^2-66x}{x^2+6x-16}-(x-8888)$$

$$ \begin{aligned} \frac{x+1}{x+8} &= \frac{x^2-66x}{x^2+6x-16}-(x-8888)&& \text{multiply ALL terms by } \color{blue}{ (x+8)(x^2+6x-16) }. \\[1 em](x+8)(x^2+6x-16)\frac{x+1}{x+8} &= (x+8)(x^2+6x-16)\frac{x^2-66x}{x^2+6x-16}-(x+8)(x^2+6x-16)x-(x+8)(x^2+6x-16)\cdot8888&& \text{cancel out the denominators} \\[1 em]x^3+7x^2-10x-16 &= x^3-58x^2-528x-(x^4+14x^3+32x^2-128x)-(8888x^3+124432x^2+284416x-1137664)&& \text{simplify right side} \\[1 em]x^3+7x^2-10x-16 &= x^3-58x^2-528x-x^4-14x^3-32x^2+128x-(8888x^3+124432x^2+284416x-1137664)&& \\[1 em]x^3+7x^2-10x-16 &= -x^4-13x^3-90x^2-400x-(8888x^3+124432x^2+284416x-1137664)&& \\[1 em]x^3+7x^2-10x-16 &= -x^4-13x^3-90x^2-400x-8888x^3-124432x^2-284416x+1137664&& \\[1 em]x^3+7x^2-10x-16 &= -x^4-8901x^3-124522x^2-284816x+1137664&& \text{move all terms to the left hand side } \\[1 em]x^3+7x^2-10x-16+x^4+8901x^3+124522x^2+284816x-1137664 &= 0&& \text{simplify left side} \\[1 em]x^4+8902x^3+124529x^2+284806x-1137680 &= 0&& \\[1 em] \end{aligned} $$

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