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The problem asks to find the expression for the profit \( P(x) \), which is given as the difference between the revenue \( R(x) \) and the cost \( C(x) \). The formula is:

\[
P(x) = R(x) - C(x)
\]

We are provided with the following expressions for revenue and cost:
- \( R(x) = 2x^4 - 3x^3 + 2x - 1 \)
- \( C(x) = x^4 - x^2 + 2x + 3 \)

To solve this, we subtract \( C(x) \) from \( R(x) \):

\[
P(x) = (2x^4 - 3x^3 + 2x - 1) - (x^4 - x^2 + 2x + 3)
\]

Distribute the negative sign to the terms in \( C(x) \):

\[
P(x) = 2x^4 - 3x^3 + 2x - 1 - x^4 + x^2 - 2x - 3
\]

Now, combine like terms:

- \( 2x^4 - x^4 = x^4 \)
- \( -3x^3 \) (there is no other \( x^3 \)-term, so it remains the same)
- \( x^2 \) (there is no \( x^2 \)-term in \( R(x) \), so it remains the same)
- \( 2x - 2x = 0 \)
- \( -1 - 3 = -4 \)

Thus, the simplified expression for \( P(x) \) is:

\[
P(x) = x^4 - 3x^3 + x^2 - 4
\]

This matches the third option in the list: \( x^4 - 3x^3 + x^2 - 4 \).

### Would you like further details on this process or have any questions?

### Related Questions:
1. How would the result change if there was an additional term in the cost function?
2. What happens if both the revenue and cost functions are linear?
3. How do you find the maximum profit for a given revenue and cost function?
4. Can you calculate the break-even point from these expressions?
5. How do higher-degree polynomials in cost and revenue affect profit analysis?

### Tip:
When combining like terms, always align corresponding powers of \( x \) to avoid missing terms in the simplification.

Profit, P(x), is the difference between revenue, R(x), and cost, C(x), so P(x) = R(x) - C(x). Which expression represents P(x), if R(x) = 2x^4 - 3x^3 + 2x - 1 and C(x) = x^4 - x^2 + 2x + 3?

Learn how to calculate the profit function P(x) by subtracting the cost polynomial C(x) from the revenue polynomial R(x). Given R(x) = 2x^4 - 3x^3 + 2x - 1 and C(x) = x^4 - x^2 + 2x + 3, this solution involves combining like terms for accurate results. Suitable for students in Grades 9-11.

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