The doorway into a room is 4 feet wide and 8 feet high. what is the length of the longest rectangular panel that can be taken through this doorway diagonally

The size of the doorway places no restriction on the length of an object that can be taken through it. Usually, the length is limited by the walls that are present on either side of the doorway (the hall or room size).

Perhaps you want to know the diagonal length of the opening. The Pythagorean theorem tells you it is
   √((4 ft)² +(8 ft)²) = 4√5 ft ≈ 8.944 ft

Remark
This is a Pythagorean theorem problem. You need only apply a^2 + b^2 = c^2 to get the answer.

Formula
a^2 + b^2 = c^2

Givens
a = 4
b = 8

Sub and solve
a^2 + b^2 = c^2
4^2 + 8^2 = c^2
16 + 64 = c^2
80 = c^2
c = sqrt(80)
c = sqrt(2 * 2 * 2 * 2 * 5) out 4 twos 2 can be taken outside the brackets.
c = 4*sqrt(5) is the longest piece that can be taken through the door.
c = 8 feet + 0.944 of a foot
c = 8 feet + 11.33 inches.