Question

Which expression is equal to left-parenthesis StartFraction 3 x plus 2 over 4 x EndFraction right-parenthesis divided by left-parenthesis StartFraction x plus 5 over 8 EndFraction right-parenthesis?

Answer options with 4 options
A.
StartFraction 3 x squared plus 10 over 32 x EndFraction

B.
StartFraction 3 x squared plus 17 x plus 10 over 32 x EndFraction

C.
StartFraction 6 x plus 4 over x squared plus 5 EndFraction

D.
StartFraction 6 x plus 4 over x squared plus 5 x EndFraction

Answer 1

To simplify the expression \(\left(\frac{3x + 2}{4x}\right) \div \left(\frac{x + 5}{8}\right)\), we can multiply by the reciprocal of the second fraction:

\[
\frac{3x + 2}{4x} \times \frac{8}{x + 5}
\]

Now we can multiply the numerators and the denominators:

\[
= \frac{(3x + 2) \cdot 8}{4x \cdot (x + 5)}
\]

Simplifying the numerator:

\[
= \frac{24x + 16}{4x(x + 5)}
\]

Now we need to simplify the denominator:

\[
= 4x^2 + 20x
\]

Now we have:

\[
= \frac{24x + 16}{4x^2 + 20x}
\]

Next, we can factor the numerator and the denominator:

From the numerator:

\[
= \frac{8(3x + 2)}{4x(x + 5)}
\]

Now, let's factor the denominator:

\[
4x^2 + 20x = 4x(x + 5)
\]

So we have:

\[
= \frac{8(3x + 2)}{4x(x + 5)}
\]

Now, we can reduce the fraction:

\[
= \frac{2(3x + 2)}{x(x + 5)}
\]

Distributing in the numerator gives:

\[
= \frac{6x + 4}{x(x + 5)}
\]

Thus, we obtain:

\[
= \frac{6x + 4}{x^2 + 5x}
\]

Now, we compare this with the given options:

A. \(\frac{3x^2 + 10}{32x}\)
B. \(\frac{3x^2 + 17x + 10}{32x}\)
C. \(\frac{6x + 4}{x^2 + 5}\)
D. \(\frac{6x + 4}{x^2 + 5x}\)

The simplified form \(\frac{6x + 4}{x^2 + 5x}\) matches option D.

So the correct answer is:

**D. \(\frac{6x + 4}{x^2 + 5x}\)**