To divide the given rational expressions, follow these steps:
1. Write the division as a multiplication by the reciprocal:
[tex]\[
\frac{4c - 10}{c + 2} \div \frac{6c - 15}{4c + 8} = \frac{4c - 10}{c + 2} \times \frac{4c + 8}{6c - 15}
\][/tex]
2. Simplify each expression if possible:
- For the fraction [tex]\(\frac{4c - 10}{c + 2}\)[/tex]: Check if the numerator and denominator have common factors.
[tex]\[
4c - 10 = 2(2c - 5)
\][/tex]
Thus,
[tex]\[
\frac{4c - 10}{c + 2} = \frac{2(2c - 5)}{c + 2}
\][/tex]
- For the fraction [tex]\(\frac{4c + 8}{6c - 15}\)[/tex]: Similarly, check for common factors in the numerator and denominator.
[tex]\[
4c + 8 = 4(c + 2)
\][/tex]
[tex]\[
6c - 15 = 3(2c - 5)
\][/tex]
Thus,
[tex]\[
\frac{4c + 8}{6c - 15} = \frac{4(c + 2)}{3(2c - 5)}
\][/tex]
3. Rewrite the division problem using the simplified expressions and multiply:
[tex]\[
\frac{2(2c - 5)}{c + 2} \times \frac{4(c + 2)}{3(2c - 5)}
\][/tex]
4. Perform the multiplication and cancel out any common factors:
- Note that [tex]\((2c - 5)\)[/tex] and [tex]\((c + 2)\)[/tex] appear in both the numerator and denominator.
[tex]\[
\frac{2(2c - 5) \times 4(c + 2)}{(c + 2) \times 3(2c - 5)}
\][/tex]
- Cancel the common factors [tex]\((2c - 5)\)[/tex] and [tex]\((c + 2)\)[/tex]:
[tex]\[
\frac{2 \times 4 \times \cancel{(2c - 5)} \times \cancel{(c + 2)}}{\cancel{(c + 2)} \times 3 \times \cancel{(2c - 5)}}
\][/tex]
- This leaves us with:
[tex]\[
\frac{8}{3}
\][/tex]
So, the simplified answer is:
[tex]\[
\boxed{\frac{8}{3}}
\][/tex]
