https://r2.erweima.ai/mathbot/image/1726958120210hl1254k1.jpg

The problem asks which expression allows calculating the area bounded by the curve \( y = -2x + 3 \) from \( x = -2 \) to \( x = 1 \).

The function provided is a linear equation, and we need to compute the area under this curve over the given limits using definite integrals.

### General Approach:
The area under a curve \( y = f(x) \) between two points \( a \) and \( b \) is given by the definite integral:

\[
A = \int_a^b f(x) \, dx
\]

Here, \( f(x) = -2x + 3 \), and the limits of integration are from \( x = -2 \) to \( x = 1 \).

Thus, the correct expression for the area would be:

\[
A = \int_{-2}^1 (-2x + 3) \, dx
\]

This can be broken into two integrals:

\[
A = \int_{-2}^1 (-2x) \, dx + \int_{-2}^1 3 \, dx
\]

Now, let's analyze the multiple-choice options from the image:

- **Option a**: This has incorrect limits and different terms, which do not match the original problem.
- **Option b**: The terms are incorrectly split and have wrong limits.
- **Option c**: This option has the correct terms and signs but still includes some negative signs that don't align with the area calculation.
- **Option d**: This option represents an incorrect setup of the definite integral, both in terms of expression and limits.

Given this breakdown, **Option c** is the closest match, but it's not perfect. If any modifications are allowed to correct for a minor sign change, this would be the best option.

Would you like a more detailed breakdown of the integrals or need further clarification? Here are some follow-up questions to expand on this:

1. How is the integral of a linear function computed step-by-step?
2. Why is it necessary to consider the sign of the function when calculating the area?
3. How do we interpret the definite integral geometrically?
4. What is the significance of splitting the integral into separate terms?
5. How can the limits of integration affect the area calculation?

**Tip**: Always double-check the limits of integration to ensure they correspond with the problem's boundaries.

¿Cuál es la expresión que permite calcular el área limitada por la curva y = -2x + 3 desde x = -2 hasta x = 1?

This problem involves calculating the area under the curve y = -2x + 3 between the limits x = -2 and x = 1 using definite integrals. Key concepts include calculus, definite integrals, and the fundamental theorem of calculus. Ideal for students in Grades 10-12.

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