Additional solutions to quadratic equations which don't match the formula answer. I'm hoping for link to some resource which can explain why the following is true.

$$ x^2 + 104x - 896 = 0 $$

Using the quadratic formula we pull a = 1, b = 104, c = 896. Putting that into the formula for the discriminant we get $ 104^2 - 4.1.896 $ which is 10816 - 3584 = 7232.

Using the quadratic formula the discriminant is 7232 and using the quadratic formula the answer is 

$$ -104 \pm \sqrt{7232} \over 2 $$

This simplfies to $ -72 \pm 4 \sqrt{113} $.

The problem I have is I have not found anything which explains why if you plug x = 8 into the equation it also balances out. What I have found out infers x = 8 being an answer to the equation but not a factor and so not a solution but I don't really get the point.

Any links that explain the distinction would help. I have an infinite number of these equations which I'm looking for integer answers to so if said link also pointed out how you can obtain the integer answers like the 8 in this case instead of the irrational provided by the quadratic formula that would be great.
The quadratic formula is $$x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}.$$
Thus in your problem, $$x = \frac{-104\pm\sqrt{104^2 - 4(-896)}}{2}
=\frac{-104 \pm\sqrt{14400}}{2} = \frac{-104 \pm 120}{2}.$$
This gives $x = 8$ or $-112$.

It seems the problem in your computation was you forgot the negative on $896$ when you plugged it into the quadratic formula.
obviously 8 is one of the solutions with no doubts at all.