Find all integers n such that the quadratic $5x^2 + nx – 13$ can be expressed as the product of two linear factors with integer coefficients. I am unsure of how to approach this problem. I have thought about using the Rational root theorem, but I am unsure if this answers the question being asked.
Using the theorem, I get $\frac{p}{q} = \pm 1, \pm 13, \pm \frac{1}{5}$, and $\pm \frac{13}{5}$ as possible roots. Then I use synthetic division and Horner's method to get a remainder of $-(n+8)$. For this to be a solution, $-(n+8)=0$, so $n = -8$. Then I could do this for $+1, +13, -13,$ etc.
Is this the correct approach to answering the original question? Original question: 

Find all integers $n$ such that the quadratic $5x^2 + nx – 13$ can be expressed as the product of two linear factors with integer coefficients.

Why would I need to have a rational root to answer the problem? Couldn't I have complex solutions where I can express $5x^2 + nx - 13$ (where n is an integer) as a product of two linear factors with integer coefficients?
I greatly appreciate any insight you could provide on this. It's been about 2 years since I've done any mathematics (a brief foray into Neuroscience turned into a longer expedition than intended) and I am longing to return to the beautiful realm of mathematics. Thanks for your time in reading through this jumbled mathematical thought!

If a quadratic polynomial $ax^2 + bx + c$ with integer coefficients, $a \neq 0$, has factorization 
$$ax^2 + bx + c = (rx + s)(tx + u) \tag{1}$$
then 
\begin{align*}
a & = rt\\
b & = ru + st\\
c & = su
\end{align*}
which can be demonstrated by substituting $0$, $1$, and $-1$ in equation 1, then solving the resulting system of equations.  Note that $ac = (rt)(su) = rstu = (ru)(st)$, that is, $b$ is the sum of two integers whose product is $ac$.
For the equation $5x^2 + nx - 13$, this means that $n$ is the sum of two integers with product $5 \cdot -13 = -65$.  Since 
\begin{align*}
-65 & = 1 \cdot -65 & -65 & = -1 \cdot 65\\
    & = 5 \cdot -13 & & = -5 \cdot 13
\end{align*}
the possible integer values for $n$ are 
\begin{align*}
1 + (-65) & = -64 & -1 + 65 & = 64\\
5 + (-13) & = -8 & -5 + 13 & = 8
\end{align*}
