If $a^2 + 4b^2 + 25c^2 + 18 = 2(a - 2b + 20 c),$ then what is the value of $(a + 2b + 5c)$ ?

4

 a² + 4b² + 25c² + 18 = 2(a - 2b + 20c)

= a² + 4b² + 25c² - 2a + 4b - 40c + 18 = 0

= (a² - 2a + 1) + (4b² + 4b + 1) + (25c² - 40c + 16) = 0

= (a - 1)² + (2b + 1)² + (5c - 4)² = 0

a - 1 = 0, a = 1

 2b + 1 = 0, b = \(\frac{-1}{2}\)

5c - 4 = 0 , c = \(\frac{4}{5}\)

so the value of $(a + 2b + 5c)$ = $(1 - 2 × \frac{1}{2} + 5 × \frac{4}{5}$) = 4