The product of two consecutive integers is 11 more than their sum. write an equation and solve to find the integers.

First of all let me point out, 4 and 5 are indeed correct HOWEVER what about -3 and -2 ? 
They are most definitely consequtive integers and they satisfy the criteria in your question 
(Check: -3*-2 = 6, -3+-2= -5, 6 - -5 = 11) 

I'd like to point out that I didn't just 'spot' these solutions, I actually used a method. Read on if you want to know. 
Let x and y be the integers we want to find. We have two bits of information: 
1) They are consequtive, in algebraic terms we write this as y-x =1 or equivalently y=x+1 
(note we choose y bigger than x without loss of generality here) 

2)Their product is 11 more than their sum, again in algebraic terms: xy = x+y+11 

Now we can solve these two simultaneous equations as follows 
from 1) substitute y=x+1 into our second equation to get 
x(x+1) = x + (x+1) + 11 and rearrange to get 
x^2-x-12=0 this factorises to (x-4)(x+3)=0 
so our solutions for x are x=4 or x=-3 
now substitute these x values into y=x+1 to get y=5 or y=-2 
So we have two solutions to this problem 4 and 5 OR -3 and -2 

Hope that helps! :)