Let the three consecutive even integers be:

• $x$ (the smallest)
• $x + 2$ (the middle)
• $x + 4$ (the largest)

According to the problem, the sum of the smallest number and twice the middle number is 20 more than the largest number. Mathematically, this can be written as:

$x + 2(x + 2) = (x + 4) + 20$

Step 1: Expand and simplify the equation

$x + 2x + 4 = x + 4 + 20$

$3x + 4 = x + 24$

Step 2: Subtract $x$ from both sides

$3x - x + 4 = 24$

$2x + 4 = 24$

Step 3: Subtract 4 from both sides

$2x = 20$

Step 4: Divide both sides by 2

$x = 10$

Step 5: Find the three consecutive even integers

• The smallest number is $x = 10$
• The middle number is $x + 2 = 12$
• The largest number is $x + 4 = 14$

Thus, the three consecutive even integers are $10$, $12$, and $14$.

Verification:

The sum of the smallest number and twice the middle number: $10 + 2(12) = 10 + 24 = 34$

The largest number plus 20: $14 + 20 = 34$

Both sides are equal, so the solution is correct.

Would you like any more details or have any questions? Here are some related questions:

1. What is the general formula for finding consecutive even integers?
2. How would the equation change if the integers were consecutive odd numbers?
3. Can this problem be solved using a different method, such as substitution?
4. What are the properties of even numbers that are important in solving this problem?
5. How would the solution change if we added more conditions to the problem?

Tip: Always verify your solution by substituting the values back into the original equation to ensure accuracy.