So the 3 consecutive integers can be denoted as n,n+1 and n+2.

So turn the statement from English to math.

n(n+2)=17+7(n+1). Now simplify and solve

n^{2}+2n=17+7n+7

now bring everything to 1 side.

n^{2}-5n-24=0. The factors of -24 that sum to a -5 are -8 and 3 so this can be factored into

(n-8)(n+3)=0

n=8 or -3. So now since they say integers we need to examine the possibilities of the different integers.

If n is 8 then the numbers are 8,9,10. So let's check the statement. 8(10) compared to 7(9)+17. This checks out.

Now let's check the other possibilities.

-3,-2,-1.

(-3)(-1) compared to 7(-2)+17

3=3 and so there are 2 sets of solutions to this problem.

-3,-2,-1 and 8,9,10