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The question is asking for the area of Dylan's room, where the length and width are represented by the polynomials \( (x^2 - 2x + 8) \) and \( (2x^2 + 5x - 7) \). To find the area \( A \), we need to multiply these two expressions.

### Step 1: Set up the product of the polynomials
We are multiplying:
\[
(x^2 - 2x + 8) \cdot (2x^2 + 5x - 7)
\]

### Step 2: Apply the distributive property (FOIL method or expansion)
Multiply each term in the first polynomial by each term in the second polynomial:

\[
A = x^2 \cdot (2x^2 + 5x - 7) + (-2x) \cdot (2x^2 + 5x - 7) + 8 \cdot (2x^2 + 5x - 7)
\]

### Step 3: Expand each term

1. \( x^2 \cdot (2x^2 + 5x - 7) = 2x^4 + 5x^3 - 7x^2 \)
2. \( -2x \cdot (2x^2 + 5x - 7) = -4x^3 - 10x^2 + 14x \)
3. \( 8 \cdot (2x^2 + 5x - 7) = 16x^2 + 40x - 56 \)

### Step 4: Combine like terms
Now, sum up all the terms:
\[
A = (2x^4) + (5x^3 - 4x^3) + (-7x^2 - 10x^2 + 16x^2) + (14x + 40x) - 56
\]

Simplify:
\[
A = 2x^4 + (5x^3 - 4x^3) + (-7x^2 - 10x^2 + 16x^2) + (14x + 40x) - 56
\]
\[
A = 2x^4 + x^3 - x^2 + 54x - 56
\]

### Step 5: Match the answer
The expression that matches this result is:
\[
2x^4 + x^3 - x^2 + 54x - 56
\]

So, the correct answer is the third option:
\[
\boxed{2x^4 + x^3 - x^2 + 54x - 56}
\]

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Do you want more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How do you multiply polynomials step by step?
2. What happens when two binomials are multiplied?
3. Can this method of polynomial multiplication be applied to more than two terms in each factor?
4. How can you factor the resulting polynomial if needed?
5. How would the solution change if the polynomials were cubic or higher?

**Tip**: Always group like terms after expanding polynomials to simplify the expression correctly.

Dylan uses the expressions (x^2 − 2x + 8) and (2x^2 + 5x − 7) to represent the length and width of his bedroom. Which expression represents the area (A) of Dylan's room?

Learn how to find the area of a room where the length and width are represented by polynomials. This problem covers key algebraic concepts, including polynomial multiplication and the distributive property. Suitable for students in Grades 8-10.

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