$$x^{3}-8x^{2}+19x-12\div(x-1)$$
$\frac{x^{4}-9x^{3}+27x^{2}-19x-12}{x-1}$
$$\frac{\left(x^{3}-8x^{2}+19x\right)\left(x-1\right)}{x-1}-\frac{12}{x-1}$$
$$\frac{\left(x^{3}-8x^{2}+19x\right)\left(x-1\right)-12}{x-1}$$
$$\frac{x^{4}-x^{3}-8x^{3}+8x^{2}+19x^{2}-19x-12}{x-1}$$
$$\frac{x^{4}-9x^{3}+27x^{2}-19x-12}{x-1}$$
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$\frac{3x^{4}-22x^{3}+54x^{2}-54x+31}{\left(x-1\right)^{2}}$
