Let's define the three consecutive even integers:

• Let the smallest integer be $x$.
• The next consecutive even integer is $x + 2$.
• The largest integer is $x + 4$.

The problem states that twice the smallest integer is eight more than the largest. We can express this as an equation:

$2x = (x + 4) + 8$

Now, let's solve the equation:

1. Simplify the right-hand side:

$2x = x + 12$

2. Subtract $x$ from both sides:

$x = 12$

Thus, the smallest integer is $x = 12$, the next integer is $x + 2 = 14$, and the largest integer is $x + 4 = 16$.

So, the three consecutive even integers are 12, 14, and 16.

Would you like more details on the steps, or do you have any other questions?

Here are five related questions to explore:

1. How would the solution change if the integers were consecutive odd integers instead?
2. How can you set up equations for other word problems involving consecutive numbers?
3. What is the relationship between consecutive even or odd integers and arithmetic sequences?
4. What happens if we change the "eight more than the largest" to a different number?
5. How can you use similar steps to solve problems with more than three consecutive integers?

Tip: When working with consecutive numbers, always express the terms in terms of a single variable, then set up an equation to solve for that variable.