The given expression is:

\[
(px + q) \left( \frac{r}{x} + s \right)
\]

To simplify this, we will use the distributive property to expand the terms. Let's go step by step:

1. Distribute \((px + q)\) to both terms inside the brackets \(\left( \frac{r}{x} + s \right)\):

\[
(px + q) \times \frac{r}{x} + (px + q) \times s
\]

2. Now, distribute each term:

- For the first part:
  \[
  (px + q) \times \frac{r}{x} = px \times \frac{r}{x} + q \times \frac{r}{x}
  \]
  Simplifying:
  \[
  px \times \frac{r}{x} = r \quad \text{and} \quad q \times \frac{r}{x} = \frac{qr}{x}
  \]
  So the first part becomes:
  \[
  r + \frac{qr}{x}
  \]

- For the second part:
  \[
  (px + q) \times s = px \times s + q \times s = psx + qs
  \]

3. Combine all terms:
\[
r + \frac{qr}{x} + psx + qs
\]

Thus, the expanded form of \((px + q) \left( \frac{r}{x} + s \right)\) is:

\[
r + \frac{qr}{x} + psx + qs
\]

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. How would you simplify \((2x + 3) \left( \frac{5}{x} + 4 \right)\)?
2. What happens if \(p = 0\) in the given expression?
3. Can you factor \((psx + qs)\) out of the final expression?
4. How would the expression change if \(x\) is negative?
5. How do you handle fractional powers of \(x\) in similar expansions?

**Tip**: When expanding terms, always distribute each term carefully to avoid mistakes in combining like terms.

Explore how to simplify the expression (px + q) [r/x + s] using the distributive property. This detailed step-by-step guide covers rational expressions and algebraic distribution, ideal for students in Grades 9-11.

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