https://file.aiquickdraw.com/mathbot/file/1730789602467k76vzv54.jpg

To help you factor the given expression, let’s analyze it. The expression in the image is:

\[
\frac{\frac{3}{x^5} + 5x^5 - \frac{3}{x^5} - 15}{\frac{1}{x^5} + 5}
\]

### Step 1: Simplify the Expression

Observe that in the numerator, there are terms \(\frac{3}{x^5}\) and \(-\frac{3}{x^5}\), which will cancel each other out. Thus, we are left with:

\[
\frac{5x^5 - 15}{\frac{1}{x^5} + 5}
\]

### Step 2: Factor Out Terms

Now, in the numerator \(5x^5 - 15\), we can factor out a 5:

\[
\frac{5(x^5 - 3)}{\frac{1}{x^5} + 5}
\]

### Step 3: Rewrite the Denominator

In the denominator, \(\frac{1}{x^5} + 5\), let’s rewrite it by combining terms over a common denominator:

\[
\frac{1 + 5x^5}{x^5}
\]

So the entire expression now becomes:

\[
\frac{5(x^5 - 3)}{\frac{1 + 5x^5}{x^5}}
\]

### Step 4: Simplify the Fraction

To simplify further, multiply the numerator by \(x^5\) to eliminate the denominator in the fraction:

\[
= \frac{5x^5(x^5 - 3)}{1 + 5x^5}
\]

Thus, the factored form of the expression is:

\[
\frac{5x^5(x^5 - 3)}{1 + 5x^5}
\]

Would you like further details on any step? 

Here are some related questions to consider:
1. What other methods can simplify complex rational expressions?
2. How do you determine the lowest common denominator in fractions?
3. How does factoring help in simplifying algebraic expressions?
4. What are common errors to avoid in simplifying rational expressions?
5. How does the cancellation of terms affect the factorization process?

**Tip:** Always look for terms that can be factored out at an early stage to make complex expressions simpler.

Factor \(\frac{\frac{3}{x^5} + 5x^5 - \frac{3}{x^5} - 15}{\frac{1}{x^5} + 5}\)

Learn how to factor the complex rational expression \(\frac{\frac{3}{x^5} + 5x^5 - \frac{3}{x^5} - 15}{\frac{1}{x^5} + 5}\) through simplification and factoring techniques. This solution covers key algebraic concepts such as simplifying rational expressions and factoring polynomials. Suitable for students in Grades 10-12.

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