Let's start by dividing our first term of the polynomial, 12x^4, by the first term of the divisor, which is 3x^2. We get 4x^2. This is the first term of our quotient.
Then, we multiply the divisor 3x^2 + 1 by our first term of the quotient, 4x^2, we get 12x^4 + 4x^2 and subtract that from the original polynomial. The resulting polynomial after the subtraction is: 6x^3 + 13x^2 - 3x - 6.
Next, we divide the first term of the resulting polynomial, 6x^3, by the first term of the divisor, 3x^2, and get 2x. So, our second term of the quotient is 2x.
We again multiply the divisor 3x^2 + 1 by this new term of the quotient, 2x, to get 6x^3 + 2x, and subtract that from the remaining polynomial after the first subtraction. The result after the second subtraction is: 7x^2 - 5x - 6.
Next, we divide the first term of the new polynomial, 7x^2, by the first term of the divisor, 3x^2, and get 7/3. This gives our third term of the quotient.
If we multiply the divisor 3x^2 + 1 by this third term of the quotient, we have 7x^2 + 7/3, and if we subtract this from the current polynomial, we have -9x - 19/3.
So, we see that the degree of this polynomial (-9x - 19/3) is less than the degree of the divisor (3x^2 + 1), which means that our division process stops here. The equation is in the form
```
q(x) + b(x) = 4x^2 + 2x + 7/3 + (-9x - 19/3)/(3x^2 + 1)
```
So, the quotient and the remainder are:
q(x) = 4x^2 + 2x + 7/3
b(x) = (-9x - 19/3)
This is the same as the result we obtained from the Python code.