1. **State the problem:** Find the value of $c$ in the equation $2x + 5y = c$ for the line passing through points $(-1, 2)$ and $(4, 0)$. 2. **Formula and rule:** The equation of a line in the form $Ax + By = C$ must satisfy the coordinates of any point on the line. So, substituting either point into the equation will allow us to solve for $c$. 3. **Substitute point $(-1, 2)$:** $$2(-1) + 5(2) = c$$ $$-2 + 10 = c$$ $$c = 8$$ 4. **Verify with point $(4, 0)$:** $$2(4) + 5(0) = c$$ $$8 + 0 = c$$ $$c = 8$$ 5. Both points give the same value $c=8$, confirming the solution. **Final answer:** $$\boxed{8}$$