Let x be the first even integer. So the other two numbers can be obtained as
x+2 and x+4, where x+2 is the second even integer and x+4 is the third even integer.
Now, we want to translate the given property into an equation. We are told "three times the second", so this translates to the expression

now, we are told that this is six more than the sum of the first and the third. That is, we sum the first and the third and then add 6. This would give us this quantity. So we have

if we expand the factors on the right, we get

now, we can subtract 6 on both sides, so we get

If we operate on the left, we get

if we subtract 2x on both sides, we get

so we have that the first integer is 4. THis means that the other two integers are 6 and 8.