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: $\boxed{2x^4 + x^3 - x^2 + 54x - 56}$

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.