To determine the quotient of the rational expression [tex]\(\frac{3x + 1}{x + 8} \div \frac{x - 7}{5x}\)[/tex], we proceed with the following steps:
1. Rewrite the Division as Multiplication:
Division of fractions can be converted into multiplication by the reciprocal. Therefore:
[tex]\[
\frac{3 x + 1}{x + 8} \div \frac{x - 7}{5 x} = \frac{3 x + 1}{x + 8} \times \frac{5 x}{x - 7}
\][/tex]
2. Multiply the Numerators Together:
Multiply the numerators of the two rational expressions:
[tex]\[
(3 x + 1) \times 5 x = 5 x (3 x + 1)
\][/tex]
Simplifying the product:
[tex]\[
5 x (3 x + 1) = 15 x^2 + 5 x
\][/tex]
3. Multiply the Denominators Together:
Similarly, multiply the denominators of the two rational expressions:
[tex]\[
(x + 8) \times (x - 7)
\][/tex]
Expanding the product using the distributive property (FOIL):
[tex]\[
(x + 8)(x - 7) = x^2 - 7x + 8x - 56 = x^2 + x - 56
\][/tex]
4. Form the New Rational Expression:
Place the simplified numerator and denominator into a single fraction:
[tex]\[
\frac{15 x^2 + 5 x}{x^2 + x - 56}
\][/tex]
Therefore, the quotient of the given rational expression is:
[tex]\[
\boxed{\frac{15 x^2 + 5 x}{x^2 + x - 56}}
\][/tex]
The answer corresponds to option D.
