Let the two consecutive integers be $x$ and $x + 1$, where $x$ is the lesser of the two.

We are given that four times the lesser of the two integers is 12 less than twice the greater. This can be written as:

$4x = 2(x + 1) - 12$

Now, solve this equation step by step:

1. Expand the right-hand side:

$4x = 2x + 2 - 12$

2. Simplify the right-hand side:

$4x = 2x - 10$

3. Subtract $2x$ from both sides:

$4x - 2x = -10$

$2x = -10$

4. Divide both sides by 2:

$x = -5$

So, the lesser integer is $x = -5$, and the greater integer is $x + 1 = -4$.

Therefore, the two consecutive integers are $-5$ and $-4$.

Would you like more details or have any questions about this? Here are some related questions for practice:

1. If three times the greater of two consecutive integers is equal to the sum of the lesser integer and 10, what are the integers?
2. What are two consecutive integers such that five times the lesser is 20 more than the greater?
3. If the square of the greater of two consecutive integers is 9 more than the square of the lesser, what are the integers?
4. Find two consecutive even integers such that four times the lesser equals six less than the greater.
5. What are two consecutive integers such that twice the lesser is equal to one more than the greater?

Tip: In problems involving consecutive integers, using algebraic expressions like $x$ and $x + 1$ (for consecutive numbers) or $x$ and $x + 2$ (for consecutive even or odd numbers) simplifies setting up equations.