Answer :

To divide the polynomial [tex]\(12x^3 - 6x^2 - 3x\)[/tex] by [tex]\(-3x\)[/tex], we perform polynomial division step by step.

1. First Term Division:

We'll start by dividing the leading term of the polynomial by the leading term of the divisor:
[tex]\[ \frac{12x^3}{-3x} = -4x^2 \][/tex]

2. Second Term Division:

Next, we divide the second term of the polynomial by the divisor:
[tex]\[ \frac{-6x^2}{-3x} = 2x \][/tex]

3. Third Term Division:

Finally, we divide the third term of the polynomial by the divisor:
[tex]\[ \frac{-3x}{-3x} = 1 \][/tex]

4. Combining Terms:

Now we combine these results to obtain the quotient:
[tex]\[ -4x^2 + 2x + 1 \][/tex]

Therefore, the quotient obtained by dividing [tex]\(12x^3 - 6x^2 - 3x\)[/tex] by [tex]\(-3x\)[/tex] is:

[tex]\[ \boxed{-4x^2 + 2x + 1} \][/tex]

So, among the given choices, the correct answer is:

[tex]\[ -4x^2 + 2x + 1 \][/tex]