To divide the given rational expressions, follow these steps:
1. Rewrite the Division as Multiplication:
When dividing two rational expressions, you can multiply by the reciprocal of the second expression:
[tex]\[
\frac{\left(\frac{3 x + 8}{x^2 + 2 x - 15}\right)}{\left(\frac{3 x^2 - x - 24}{2 x^2 + 15 x + 25}\right)} = \frac{3 x + 8}{x^2 + 2 x - 15} \times \frac{2 x^2 + 15 x + 25}{3 x^2 - x - 24}
\][/tex]
2. Factor the Denominators if Possible:
[tex]\[
x^2 + 2 x - 15 = (x + 5)(x - 3)
\][/tex]
[tex]\[
2 x^2 + 15 x + 25 = (2 x + 5)(x + 5)
\][/tex]
3. Replace Factored Expressions:
Substitute the factored forms into the expression:
[tex]\[
\frac{3 x + 8}{(x + 5)(x - 3)} \times \frac{(2 x + 5)(x + 5)}{3 x^2 - x - 24}
\][/tex]
4. Factor Numerator and Denominator if Possible:
Let’s assume [tex]\(3 x^2 - x - 24\)[/tex] can be factored:
[tex]\[
3 x^2 - x - 24 = (3 x + 8)(x - 3)
\][/tex]
5. Substitute back into the expression:
[tex]\[
\frac{3 x + 8}{(x + 5)(x - 3)} \times \frac{(2 x + 5)(x + 5)}{(3 x + 8)(x - 3)}
\][/tex]
6. Cancel Common Factors:
Cancel out the common factors [tex]\((3 x + 8)\)[/tex], [tex]\((x + 5)\)[/tex], and [tex]\((x - 3)\)[/tex] from the numerator and the denominator:
[tex]\[
\frac{(2 x + 5)}{(x - 3)}
\][/tex]
8. Simplified Result:
The final result of the division is:
[tex]\[
\frac{2 x + 5}{x^2 - 6x + 9}
\][/tex]
Therefore, the completely simplified rational expression is:
[tex]\[
\boxed{\frac{2 x + 5}{x^2 - 6 x + 9}}
\][/tex]
