Normals are drawn to the parabola y² = 4x from the point (5, 0)…

Step 1. The cubic. $0 = 5m - 2m - m^3$, so $m^3 - 3m = 0$ and $m(m^2 - 3) = 0$. The roots are $m = 0$ and $m = \pm\sqrt{3}$. That is $3$ normals.

Step 2. The feet. For $m = \pm\sqrt{3}$ the foot is $\left(m^2, -2m\right) = (3, \mp 2\sqrt{3})$. Both have $x = 3$. Check: $12 = 4 \times 3$.

Step 3. The distance. The two feet are $(3, 2\sqrt{3})$ and $(3, -2\sqrt{3})$, so the distance is $4\sqrt{3} \approx 6.93$.