To determine the area of Dylan's room, we'll use the given expressions for length and width and multiply them.
The expressions are:
- Length: [tex]\( x^2 - 2x + 8 \)[/tex]
- Width: [tex]\( 2x^2 + 5x - 7 \)[/tex]
When we multiply these two expressions together, we get:
[tex]\[
\text{Area} = (x^2 - 2x + 8)(2x^2 + 5x - 7)
\][/tex]
We need to perform the multiplication step-by-step using the distributive property, also known as the FOIL method (First, Outer, Inner, Last) for polynomials:
First terms:
[tex]\[ x^2 \cdot 2x^2 = 2x^4 \][/tex]
Outer terms:
[tex]\[ x^2 \cdot 5x = 5x^3 \][/tex]
Inner terms:
[tex]\[ x^2 \cdot (-7) = -7x^2 \][/tex]
Second term first term:
[tex]\[ -2x \cdot 2x^2 = -4x^3 \][/tex]
Second term outer term:
[tex]\[ -2x \cdot 5x = -10x^2 \][/tex]
Second term inner term:
[tex]\[ -2x \cdot (-7) = 14x \][/tex]
Last terms:
[tex]\[ 8 \cdot 2x^2 = 16x^2 \][/tex]
Last term outer term:
[tex]\[ 8 \cdot 5x = 40x \][/tex]
Last term inner term:
[tex]\[ 8 \cdot (-7) = -56 \][/tex]
Now, we sum all these individual terms:
[tex]\[
2x^4 + 5x^3 - 7x^2 - 4x^3 - 10x^2 + 14x + 16x^2 + 40x - 56
\][/tex]
Next, let's combine like terms:
[tex]\[
2x^4 + (5x^3 - 4x^3) + (-7x^2 - 10x^2 + 16x^2) + (14x + 40x) - 56
\][/tex]
This simplifies to:
[tex]\[
2x^4 + x^3 - x^2 + 54x - 56
\][/tex]
Therefore, the expression that represents the area of Dylan's room is:
[tex]\[
2x^4 + x^3 - x^2 + 54x - 56
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{2x^4 + x^3 - x^2 + 54x - 56}
\][/tex]
