To solve the problem, we need to find the quotient of two fractions involving [tex]\( x \)[/tex]:
[tex]\[
\frac{2x-3}{x} \div \frac{7}{x^2}
\][/tex]
The division of fractions can be handled by multiplying by the reciprocal of the second fraction. In other words, we have:
[tex]\[
\frac{2x-3}{x} \div \frac{7}{x^2} = \frac{2x-3}{x} \times \frac{x^2}{7}
\][/tex]
Let's perform this multiplication step-by-step:
1. Numerator Multiplication:
[tex]\[
(2x - 3) \times x^2 = (2x-3)x^2 = 2x^3 - 3x^2
\][/tex]
2. Denominator Multiplication:
[tex]\[
x \times 7 = 7x
\][/tex]
3. Putting it Together:
[tex]\[
\frac{(2x-3)x^2}{7x} = \frac{2x^3 - 3x^2}{7x}
\][/tex]
4. Simplifying the Fraction:
To simplify [tex]\(\frac{2x^3 - 3x^2}{7x}\)[/tex], divide each term in the numerator by [tex]\( x \)[/tex]:
[tex]\[
\frac{2x^3 - 3x^2}{7x} = \frac{2x^2(2x-3)}{7x(2x-3)}
\][/tex]
Simplifying further by canceling common terms:
- The numerator [tex]\( 2x^3 - 3x^2 \)[/tex] can be written as [tex]\( x(2x^2 - 3) \)[/tex].
- The denominator remains [tex]\( 7x \)[/tex].
Finally, observe that the [tex]\( x \)[/tex] in the numerator and denominator cancels each other out, leaving:
[tex]\[
= \frac{x(2x-3)}{7}
\][/tex]
Thus, the resulting simplified quotient is:
[tex]\[
\frac{x(2x-3)}{7}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{D. \frac{x(2x-3)}{7}}
\][/tex]
