To solve this problem, we need to find the area of Dylan's room given the length and width expressions. The area of a rectangle is obtained by multiplying the length by the width.
1. The length of Dylan's room is given by [tex]\( L = x^2 - 2x + 8 \)[/tex].
2. The width of Dylan's room is given by [tex]\( W = 2x^2 + 5x - 7 \)[/tex].
To find the area, we need to multiply these two expressions:
[tex]\[ \text{Area} = L \times W \][/tex]
[tex]\[ \text{Area} = (x^2 - 2x + 8) \times (2x^2 + 5x - 7) \][/tex]
Upon performing the polynomial multiplication and combining like terms, the results yield:
[tex]\[ \text{Area} = 2x^4 + x^3 - x^2 + 54x - 56 \][/tex]
Hence, the expression that correctly represents the area of Dylan's room is:
[tex]\[ 2x^4 + x^3 - x^2 + 54x - 56 \][/tex]
So, the correct answer is:
[tex]\[ 2x^4 + x^3 - x^2 + 54x - 56 \][/tex]
This matches the third option provided in your question:
[tex]\[ \boxed{2 x^4 + x^3 - x^2 + 54 x - 56} \][/tex]
