Wir haben folgendes: $$\frac{x^3+2x^2-11x-12}{x-4}=\frac{x^3+2x^2-4x^2+4x^2-11x-12}{x-4} \\ =\frac{x^3-4x^2+6x^2-11x-12}{x-4} \\ =\frac{x^3-4x^2}{x-4}+\frac{6x^2-11x-12}{x-4} \\ =\frac{x^2(x-4)}{x-4}+\frac{6x^2-11x-12}{x-4} \\ =x^2+\frac{6x^2-11x-12}{x-4} \\ =x^2+\frac{6x^2-6\cdot 4x+6\cdot 4x-11x-12}{x-4} \\ =x^2+\frac{6x^2-24x+24x-11x-12}{x-4} \\ =x^2+\frac{6x^2-24x}{x-4}+\frac{13x-12}{x-4} \\ =x^2+6x\cdot \frac{x-4}{x-4}+\frac{13x-12}{x-4} \\ =x^2+6x+\frac{13x-12}{x-4} \\ =x^2+6x+\frac{13x-4\cdot 13+4\cdot 13-12}{x-4} \\ =x^2+6x+\frac{13x-4\cdot 13}{x-4}+\frac{4\cdot 13-12}{x-4} \\ =x^2+6x+13\frac{x-4}{x-4}+\frac{40}{x-4} \\ =x^2+6x+13+\frac{40}{x-4}$$