Let's define the four consecutive even integers. We can express them in terms of a variable:

• Let the first integer be $x$.
• The second integer will then be $x + 2$.
• The third integer will be $x + 4$.
• The fourth integer will be $x + 6$.

We are told that the sum of these four integers is $-276$. Therefore, we can set up the equation:

$x + (x + 2) + (x + 4) + (x + 6) = -276$

Simplifying the left-hand side:

$4x + 12 = -276$

Next, subtract 12 from both sides:

$4x = -276 - 12$

$4x = -288$

Now, divide both sides by 4:

$x = \frac{-288}{4} = -72$

So, the least integer is $\boxed{-72}$.

Verification:

The four integers are:

• $-72$
• $-70$
• $-68$
• $-66$

Their sum is:

$-72 + (-70) + (-68) + (-66) = -276$

This confirms that the solution is correct.

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

1. How do you solve problems involving consecutive integers?
2. What happens if the integers are odd instead of even?
3. How would the equation change if the sum was a positive number instead of negative?
4. Can you generalize this method for any number of consecutive integers?
5. What if the integers had a different common difference, like 3 instead of 2?

Tip: When dealing with consecutive numbers, always express them in terms of a variable and use the relationship between them to set up an equation.