To determine which expression is equivalent to [tex]\(\frac{12 x(x-3)(x+5)}{30 x(x+3)(x+5)}\)[/tex], we can simplify the given expression step-by-step.
1. Factorization and simplification:
[tex]\[
\frac{12 x (x-3)(x+5)}{30 x (x+3)(x+5)}
\][/tex]
2. Cancel the common factors:
- The factor [tex]\(x\)[/tex] appears in both the numerator and the denominator.
- The factor [tex]\((x+5)\)[/tex] also appears in both the numerator and the denominator.
Thus, we can cancel out these common factors:
[tex]\[
\frac{12 (x-3)}{30 (x+3)}
\][/tex]
3. Simplify the constants:
We now have:
[tex]\[
\frac{12 (x-3)}{30 (x+3)}
\][/tex]
Simplify the fraction [tex]\(\frac{12}{30}\)[/tex]:
[tex]\[
\frac{12}{30} = \frac{2}{5}
\][/tex]
So, the expression simplifies to:
[tex]\[
\frac{2 (x-3)}{5 (x+3)}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{2 (x-3)}{5 (x+3)}
\][/tex]
Now, let's compare this simplified expression with the given options:
1. [tex]\(-\frac{2 x^3}{5}\)[/tex] - This expression does not match our simplified form.
2. [tex]\(-\frac{x^3}{18}\)[/tex] - This expression is also not a match.
3. [tex]\(\frac{x^3(x-3)}{18(x+3)}\)[/tex] - This expression does not match our simplified form.
4. [tex]\(\frac{2 x^3(x-3)}{5(x+3)}\)[/tex] - This expression does not match our simplified form.
None of the given options match the simplified expression.
It appears there might be a misunderstanding or miscommunication regarding the conversion or equivalence. If we reconsider the simplified form in relation to any transformations, none of the provided options match directly.
However, if we adhere strictly to the simplifications and comparisons made step-by-step, the provided options don't directly correspond. It's crucial to ensure that we are interpreting correctly without addition other factors or transformations beyond simple algebraic simplification.
Given our correct algebraic simplification, our simplified form was:
[tex]\(\boxed{\frac{2 (x-3)}{5 (x+3)}}\)[/tex]
This does not align directly with provided options, none are essentially equivalent. Hence none of listed given expressions match up fully correct.
