To determine the area of Dylan's bedroom, we need to multiply the expressions representing the length and width. Let's break it down step-by-step:
1. Identify the expressions:
- Length of the room: [tex]\( x^2 - 2x + 8 \)[/tex]
- Width of the room: [tex]\( 2x^2 + 5x - 7 \)[/tex]
2. Multiply the expressions:
To find the area, we use the distributive property (also known as the FOIL method for binomials) to multiply these polynomials:
[tex]\[
\begin{aligned}
& (x^2 - 2x + 8)(2x^2 + 5x - 7) \\
& = x^2 \cdot (2x^2 + 5x - 7)
- 2x \cdot (2x^2 + 5x - 7)
+ 8 \cdot (2x^2 + 5x - 7) \\
&= (x^2 \cdot 2x^2) + (x^2 \cdot 5x) + (x^2 \cdot -7) \\
&\quad - (2x \cdot 2x^2) - (2x \cdot 5x) - (2x \cdot -7) \\
&\quad + (8 \cdot 2x^2) + (8 \cdot 5x) + (8 \cdot -7) \\
&= 2x^4 + 5x^3 - 7x^2 - 4x^3 - 10x^2 + 14x + 16x^2 + 40x - 56
\end{aligned}
\][/tex]
3. Combine like terms:
Now we combine the terms with the same degree:
[tex]\[
\begin{aligned}
& 2x^4 + (5x^3 - 4x^3) + (-7x^2 - 10x^2 + 16x^2) + (14x + 40x) - 56 \\
& = 2x^4 + x^3 - x^2 + 54x - 56
\end{aligned}
\][/tex]
4. Compare with options given:
After multiplying and simplifying the expression, the resulting polynomial is:
[tex]\[
2x^4 + x^3 - x^2 + 54x - 56
\][/tex]
By examining the given options, we find that this expression matches exactly with one of the options provided:
[tex]\[
\boxed{2x^4 + x^3 - x^2 + 54x - 56}
\][/tex]
Therefore, the expression that represents the area of Dylan's room is [tex]\(2x^4 + x^3 - x^2 + 54x - 56\)[/tex].
