let the 3 consecutive even integers be 2x-2, 2x, 2x+2 where x is integer

2x-2 < 2x < 2x+2

smallest: 2x-2

middle: 2x

largest: 2x+2

five times the sum of the smallest and largest integers is the same as the square of the middle number

5(2x - 2 + 2x + 2) = (2x)^2

20x = 4x^2

4x^2 - 20x = 0

by solving the quadratic equations we find 2 solutions for x

x = 0

the three consecutive even integers are: -2, 0, 2

x = 5

the three consecutive even integers are: 8, 10, 12