To find the quotient of the given rational expression:
[tex]\[
\frac{x-4}{3 x^2} \div \frac{12 x+1}{x+3}
\][/tex]
we can follow these steps:
1. Recall that dividing by a fraction is the same as multiplying by its reciprocal. So, the given expression is equivalent to:
[tex]\[
\frac{x-4}{3 x^2} \cdot \frac{x+3}{12 x+1}
\][/tex]
2. Next, we multiply the numerators together and the denominators together:
[tex]\[
\text{Numerator: } (x-4)(x+3)
\][/tex]
[tex]\[
\text{Denominator: } (3 x^2)(12 x+1)
\][/tex]
3. We can now represent the product as a single fraction:
[tex]\[
\frac{(x-4)(x+3)}{3 x^2 (12 x+1)}
\][/tex]
4. Let’s simplify the numerator and the denominator if possible.
First, expand the numerator:
[tex]\[
(x-4)(x+3) = x^2 + 3x - 4x - 12 = x^2 - x - 12
\][/tex]
Now, the denominator:
[tex]\[
3 x^2 (12 x+1) = 36 x^3 + 3 x^2
\][/tex]
So, the simplified quotient becomes:
[tex]\[
\frac{x^2 - x - 12}{36 x^3 + 3 x^2}
\][/tex]
5. Hence, the simplified form of the given rational expression is:
[tex]\[
\frac{x^2 - x - 12}{36 x^3 + 3 x^2}
\][/tex]
Therefore, the correct answer is:
C. [tex]\(\frac{x^2 - x - 12}{36 x^3 + 3 x^2}\)[/tex]
