<s>[INST] Finding the positive integer solutions $(x,y)$ of the equation $x^2+3=y(x+2)$ 
Finding the positive integer solutions $(x,y)$ of the equation
  $x^2+3=y(x+2)$

Source: Art of Problem Solving Vol. 2
Any help would be appreciated.
 [/INST] Divide the polynomial $x^2+3$ by the polynomial $x+2$. We get
$$\frac{x^2+3}{x+2}=x-2+\frac{7}{x+2}.$$
For an integer  $x$, the value of $x-2+\frac{7}{x+2}$ is an integer if and only if $x+2$ divides $7$. The only positive integer $x$ for which this is true is given by $x=5$.   
Comment: If we are interested in integer solutions, not necessarily positive, note that $x-2+\frac{7}{x+2}$ is an integer also at $x=-1$, $x=-3$, and $x=-9$.
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