Multiply.
StartFraction x minus 5 Over 5 x plus 10 EndFraction times StartFraction 4 x plus 8 Over 4 x minus 20 EndFraction
Question content area bottom
Part 1
Select the correct choice below and fill in the answer​ box(es) to complete your choice.
​(Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as​ needed.)
A.
StartFraction x minus 5 Over 5 x plus 10 EndFraction times StartFraction 4 x plus 8 Over 4 x minus 20 EndFraction
equals
  
enter your response here​, x not equals
  
enter your response here
B.
StartFraction x minus 5 Over 5 x plus 10 EndFraction times StartFraction 4 x plus 8 Over 4 x minus 20 EndFraction
equals
  
enter your response here​, no numbers must be excluded.

We are tasked with multiplying the two rational expressions:

\[
\frac{x - 5}{5x + 10} \times \frac{4x + 8}{4x - 20}
\]

### Step 1: Factor each part

First, we simplify each part by factoring.

- In the denominator of the first fraction, \(5x + 10\), factor out a common factor of 5:
  \[
  5x + 10 = 5(x + 2)
  \]
  So, the first fraction becomes:
  \[
  \frac{x - 5}{5(x + 2)}
  \]

- In the numerator of the second fraction, \(4x + 8\), factor out a common factor of 4:
  \[
  4x + 8 = 4(x + 2)
  \]
  So, the second fraction becomes:
  \[
  \frac{4(x + 2)}{4x - 20}
  \]

- In the denominator of the second fraction, \(4x - 20\), factor out a common factor of 4:
  \[
  4x - 20 = 4(x - 5)
  \]
  So, the second fraction becomes:
  \[
  \frac{4(x + 2)}{4(x - 5)}
  \]

### Step 2: Multiply the fractions

Now, we multiply the simplified fractions:

\[
\frac{x - 5}{5(x + 2)} \times \frac{4(x + 2)}{4(x - 5)}
\]

This becomes:

\[
\frac{(x - 5) \cdot 4(x + 2)}{5(x + 2) \cdot 4(x - 5)}
\]

### Step 3: Cancel common factors

We can cancel out the common factors of \(x + 2\) from the numerator and denominator, and also cancel the factor of 4:

\[
\frac{x - 5}{5(x - 5)}
\]

Now, we can cancel the common factor of \(x - 5\) (as long as \(x \neq 5\)):

\[
\frac{1}{5}
\]

### Step 4: Determine restrictions

Since we canceled \(x - 5\) and \(x + 2\), we need to exclude the values \(x = 5\) and \(x = -2\) from the domain.

### Final Answer

\[
\frac{1}{5}, \quad x \neq 5, -2
\]

### Option Selection

This corresponds to option A:
\[
\frac{1}{5}, \quad x \neq 5, -2
\]

---

Do you have any questions or would you like more details on any step?

Here are some related questions for further practice:
1. Simplify \(\frac{3x + 9}{9x + 27} \times \frac{4x + 12}{12x + 36}\).
2. Multiply \(\frac{x^2 - 4}{x^2 + 2x + 4} \times \frac{x + 2}{x - 2}\).
3. What happens if you multiply two rational expressions with no common factors?
4. Solve \(\frac{x^2 - 9}{x^2 + 6x + 9} \times \frac{x + 3}{x - 3}\).
5. What values of \(x\) must be excluded when simplifying \(\frac{2x}{x^2 - 9} \times \frac{x + 3}{x - 3}\)?

**Tip**: Always factor expressions fully before multiplying or dividing to identify any common factors that can simplify your work.

Multiply. (x - 5) / (5x + 10) * (4x + 8) / (4x - 20). Select the correct choice below and fill in the answer box(es) to complete your choice. Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed. A. (x - 5) / (5x + 10) * (4x + 8) / (4x - 20) = (response), x ≠ (response). B. (x - 5) / (5x + 10) * (4x + 8) / (4x - 20) = (response), no numbers must be excluded.

Learn how to multiply and simplify rational expressions like (x - 5) / (5x + 10) * (4x + 8) / (4x - 20). This step-by-step solution involves factoring, multiplying rational expressions, and determining domain restrictions. Suitable for students in Grades 8-10.

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