To determine the horizontal distance between Terry and the hoop, we need to find the points where the path of the ball (given by the equation \( y = -x^2 + 4x + 5 \)) intersects the height of the hoop \( y = 5 \).
We can set the two equations equal to find the values of \( x \):
\[ -x^2 + 4x + 5 = 5 \]
Subtract 5 from both sides:
\[ -x^2 + 4x = 0 \]
Factoring out \( x \):
\[ x(-x + 4) = 0 \]
Setting each factor to zero gives us:
- \( x = 0 \)
- \( -x + 4 = 0 \) ⇒ \( x = 4 \)
So, the ball is at the height of the hoop at \( x = 0 \) feet and \( x = 4 \) feet.
Terry throws the ball from \( x = 0 \) to the hoop which is at \( x = 4 \). Thus, the horizontal distance between Terry and the hoop is:
\[ 4 - 0 = 4 \text{ feet} \]
Therefore, the answer is 4 feet.