First do it in your head so you will know if and when you have the
right answer.

They both have to be about half of 44, or about 22, one of them a
little less and the other a little more.  21 and 23 will do it, but
they're odd, not even.  It looks like there is no way.  So I'd guess
there is no solution.  Now we'll do it algebraically:

First even integer = x
Next even integer = x+2

First integer + second integer = 44

                     x + (x+2) = 44
                     x + x + 2 = 44 
                        2x + 2 = 44
                            2x = 22
                             x = 21

And x+2 = 23.  But 21 and 23 are not even, so there is no solution.  Change "even" to "odd" and the solution will be 21 and 23.  
But with "even" there is no solution.

The rule is:

The sum of two consecutive even integers is NEVER divisible
by 4.

The sum of two consectutive odd integers is ALWAYS divisible
by 4.

Therefore 44 is the the sum of two consecutive ODD integers but
is NOT the sum of any two consecutive EVEN integers.

Edwin