I'll show you two ways to do it.  One way your teacher won't accept and one that
he or she will accept"


Here's the easy way that your teacher won't accept.  They have to both be
close to the square root of 224.  So we find the square root of 224 with a
calculator and get 14.966629555.  So we take the two consecutive even integers
that 14.966629555 is between and we get that 14.966629555 is between the two
consecutive integers 14 and 16. So they are one of the answers.  But another 
solution is found by taking the negatives of those, -16 and -14, since their
product is also +224.  Your teacher won't accept this way but it's a good
way to know when you've got the right answer.

Here's the way your teacher will accept:

Let x represent the smaller consecutive even integer
Then x + 2 represents the larger consecutive even integer

               x(x + 2) = 224

                x² + 2x = 224

          x² + 2x - 224 = 0

       (x - 14)(x + 16) = 0

    x - 14 = 0;                   x + 16 = 0

         x = 14;                      x = -16
       
 x+2 = 14 + 2 = 16;            x+2 = -16 + 2 = -14; 
 
So there are two solutions (14 and 16) and (-16 and -14)

Edwin