Integral of 6x^2-6ax-12x dx

1. Integrate term-by-term:

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \left(- 12 x\right)\, dx = - 12 \int x\, dx$$

      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

         $$\int x\, dx = \frac{x^{2}}{2}$$

      So, the result is: $- 6 x^{2}$

   1. Integrate term-by-term:

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- 6 a x\right)\, dx = - 6 a \int x\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x\, dx = \frac{x^{2}}{2}$$

         So, the result is: $- 3 a x^{2}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 6 x^{2}\, dx = 6 \int x^{2}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{2}\, dx = \frac{x^{3}}{3}$$

         So, the result is: $2 x^{3}$

      The result is: $- 3 a x^{2} + 2 x^{3}$

   The result is: $- 3 a x^{2} + 2 x^{3} - 6 x^{2}$

2. Now simplify:

   $$x^{2} \left(- 3 a + 2 x - 6\right)$$

3. Add the constant of integration:

   $$x^{2} \left(- 3 a + 2 x - 6\right)+ \mathrm{constant}$$

The answer is:

$$x^{2} \left(- 3 a + 2 x - 6\right)+ \mathrm{constant}$$