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The given problem defines the functions \( r(x) = 2x^3 \) and \( s(x) = x^2 \), and asks for the following:

1. \( (s + r)(x) \) — This represents the sum of \( s(x) \) and \( r(x) \).
   \[
   (s + r)(x) = s(x) + r(x) = x^2 + 2x^3
   \]

2. \( (s - r)(x) \) — This represents the difference between \( s(x) \) and \( r(x) \).
   \[
   (s - r)(x) = s(x) - r(x) = x^2 - 2x^3
   \]

3. Evaluate \( (s \cdot r)(-1) \) — This represents the product of \( s(x) \) and \( r(x) \) evaluated at \( x = -1 \).
   \[
   (s \cdot r)(x) = s(x) \cdot r(x) = (x^2) \cdot (2x^3) = 2x^5
   \]
   Evaluate at \( x = -1 \):
   \[
   (s \cdot r)(-1) = 2(-1)^5 = -2
   \]

### Summary of answers:
1. \( (s + r)(x) = x^2 + 2x^3 \)
2. \( (s - r)(x) = x^2 - 2x^3 \)
3. \( (s \cdot r)(-1) = -2 \)

Would you like more details on this? Here are 5 related questions you might explore:
1. How would the expressions change if \( r(x) = 3x^3 \)?
2. What is the derivative of \( s(x) \) and \( r(x) \)?
3. How would you calculate \( (r \cdot s)(2) \)?
4. What would \( (s + r)(x) \) look like for higher-degree polynomials?
5. Can you graph the functions \( s(x) \) and \( r(x) \)?

**Tip**: When combining functions, always look out for opportunities to factor the resulting expressions for simplification.

Write the expressions for (s + r)(x) and (s - r)(x) and evaluate (s ⋅ r)(-1) given r(x) = 2x^3 and s(x) = x^2.

Learn how to write the expressions for the sum and difference of functions r(x) = 2x^3 and s(x) = x^2, and evaluate the product at x = -1. This problem covers key algebraic concepts, including function operations, and is suitable for students in Grades 9-11.

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