Let $X$ a scheme. Why do localy finitely presented $\mathcal{O}_X$-modules are quasi-coherent? Does my explaination is correct: one has locally on an open $U$, $$ \mathcal{O}_U^m\to \mathcal{O}_U^n\to\mathcal{F}_{|U}\to 0 $$ so that $\mathcal{F}_{|U}\simeq\mathcal{O}_U^n/\text{Im}(g)$ with $g:\mathcal{O}_U^m\to\mathcal{O}_U^n$, but if $U=\text{Spec}(A)$ one has $\mathcal{O}_U^n=\widetilde{A^n}$ and I guess that the image $\text{Im}(g)=\widetilde{N}$ with $N$ a sub $A$-module of $A^n$ so that $\mathcal{F}_{|U}=\widetilde{A^n/N}$.
In general why locally generated $\mathcal{O}_X$-modules are not quasi-coherent? Here one has locally $f:\mathcal{O}_X^n\to\mathcal{F}_{|U}$ an epimorphism so that $\mathcal{F}_{|U}=\widetilde{A^n}/\text{ker}(f)$. The key point is then: image between quasi-coherent sheaf stay quasi-coherent (here $\text{Im}(g)$) but kernel of (quasi-coherent)$\to$($\mathcal{O}_X$-module) is not always quasi-coherent? Correct?