In future, please type out the problem instead of adding an attachment or picture (unless there are diagrams involved) - that allows responders to quote you and, in general, makes it easier to follow the thread.

We know:

x^2 + 5y = 49

or

5y = 49 - x^2

y = (49 - x^2)/5

(When a question asks you about a specific variable, isolating that variable is usually a good first step.)

Q: is y an integer?

What are we missing? The value of x^2. We have no info at all about x, so we don't want to make any assumptions. Let's look at the statements:

1) 1 < x < 4

If we let x=2, we get y = (49 - 4)/5 = 45/5 = 9. Is 9 an integer? YES

If we let x = 2.4799813, however, it's clear that Y will NOT work out to be an integer (I intentionally chose a number weird enough so that I'd know Y wouldn't be an integer without doing any math).

We can get both a YES and a NO answer: insufficient.

2) x^2 is an integer.

We can let x=2 again (since 2^2 is an integer) to get a YES answer.

If we let x = 0 (0^2 = 0, which is an integer), we get y = 49/5, which is NOT an integer.

We can get both a YES and a NO answer: insufficient.

Now we need to combine:

If 1<x<4 AND x^2 is an integer, there are still a number of possible values for x. Remember, nowhere does it say x must be an integer.

Following both rules, x could be:

root2
root3
root4 = 2
root5
root6
root7
...
root15

We already know that x = 2 gives us a YES answer.

If we let x = root5, we get:

y = (49 - 5)/5 = 44/5 which is NOT an integer.

So, even after combining we can get both a YES and a NO answer: insufficient, choose (E).