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]
