Alright, let's solve the given problem step by step.
We are given two rational expressions:
[tex]\[ \frac{x - 4}{2x^2} \][/tex]
and
[tex]\[ \frac{2x + 3}{x + 4} \][/tex]
And we need to find the quotient when the first expression is divided by the second. The division of two fractions can be converted to multiplication by inverting the second fraction. Therefore, we need to multiply the first expression by the reciprocal of the second expression:
[tex]\[
\frac{x - 4}{2x^2} \div \frac{2x + 3}{x + 4} = \frac{x - 4}{2x^2} \times \frac{x + 4}{2x + 3}
\][/tex]
Now, let's perform the multiplication:
[tex]\[
\frac{(x - 4)(x + 4)}{2x^2 (2x + 3)}
\][/tex]
We recognize that [tex]\((x - 4)(x + 4)\)[/tex] is a difference of squares:
[tex]\[
(x - 4)(x + 4) = x^2 - 16
\][/tex]
So our expression becomes:
[tex]\[
\frac{x^2 - 16}{2x^2 (2x + 3)}
\][/tex]
This is already the simplified form of our quotient. Now, we need to match this with one of the given options.
The choices are:
A. [tex]\(\frac{2x}{2x^2 + 2x + 3}\)[/tex]
B. [tex]\(\frac{2x^2 - 5x - 12}{2x^3 + 8x^2}\)[/tex]
C. [tex]\(\frac{x^2 - 16}{4x^3 + 6x^2}\)[/tex]
D. [tex]\(\frac{-2x - 3}{2x^2}\)[/tex]
Among these options, the one that matches:
[tex]\[
\frac{x^2 - 16}{2x^2 (2x + 3)}
\][/tex]
is:
C. [tex]\(\frac{x^2 - 16}{4x^3 + 6x^2}\)[/tex]
Because:
[tex]\[
4x^3 + 6x^2 = 2x^2 (2x + 3)
\][/tex]
Thus, the correct answer is option C.
