To determine the expression representing the area of Dylan's room, we need to multiply the length expression [tex]\((x^2 - 2x + 8)\)[/tex] by the width expression [tex]\((2x^2 + 5x - 7)\)[/tex].
First, let’s denote:
[tex]\[ L(x) = x^2 - 2x + 8 \][/tex]
[tex]\[ W(x) = 2x^2 + 5x - 7 \][/tex]
The area [tex]\(A(x)\)[/tex] can be found by multiplying these two expressions:
[tex]\[ A(x) = L(x) \cdot W(x) \][/tex]
We proceed with polynomial multiplication:
[tex]\[
\begin{aligned}
(x^2 - 2x + 8) & \cdot (2x^2 + 5x - 7) = \\
&= x^2 \cdot (2x^2 + 5x - 7) - 2x \cdot (2x^2 + 5x - 7) + 8 \cdot (2x^2 + 5x - 7) \\
&= x^2 \cdot 2x^2 + x^2 \cdot 5x + x^2 \cdot (-7) \\
&- 2x \cdot 2x^2 - 2x \cdot 5x - 2x \cdot (-7) \\
&+ 8 \cdot 2x^2 + 8 \cdot 5x + 8 \cdot (-7) \\
&= 2x^4 + 5x^3 - 7x^2 \\
&- 4x^3 - 10x^2 + 14x \\
&+ 16x^2 + 40x - 56 \\
\end{aligned}
\][/tex]
Next, we combine the like terms in the resulting polynomial:
[tex]\[
\begin{aligned}
A(x) &= 2x^4 + (5x^3 - 4x^3) + (-7x^2 - 10x^2 + 16x^2) + (14x + 40x) - 56 \\
&= 2x^4 + x^3 - x^2 + 54x - 56
\end{aligned}
\][/tex]
Thus, the expression representing the area of Dylan's room is given by:
[tex]\[ 2x^4 + x^3 - x^2 + 54x - 56 \][/tex]
From the given options, the correct expression is:
[tex]\[ 2x^4 + x^3 - x^2 + 54x - 56 \][/tex]
So the correct choice is the third one:
[tex]\[ 3. \ 2x^4 + x^3 - x^2 + 54x - 56 \][/tex]
