The area of new shape is $A=130$ m$^2$. The original square had $2$ m added to its width and $5$ m to its length, the problems asks for one of the original sides (since they're all equal of course).
How can I go about this? I figured $130$ m$^2=(L+5)(W+2)$
But doesn't seem to be coming up right.
On
So given that the original square had sides of $x$ m, the area of this new shape is $(x+5)(x+2)$ m$^2$
We can find $x$ with a standard quadratic solver, which might be the intended method of solution, but it's pretty easy to just walk back through the equations.
$$\begin{align} 130 &= (x+5)(x+2)\\ &= x^2+7x+10\\ 120 &= x^2+7x\\ &= x(x+7)\\ \end{align}$$
So - if we're going to find the answer in integers - we need $x$ and $x+7$ to be a factor pair of $120$. And $120=8\times 15$ gives us $x=8$.
Since we're talking about a measurement, the factor pair $(-15,-8)$ doesn't apply here.
It is important to understand here that in your context, L = W. Using this, you can arrive at a quadratic equation in terms of either L or W, and can thus solve the question.