jmc

Polynomial long division gives us $$\begin{array}{cccc}\text{multicolumn}2rx& +2& & \\ \text{cline}2-42x+3& 2x^2& +7x& +10\\ \text{multicolumn}2r2x^2& +3x& & \\ \text{cline}2-3\text{multicolumn}2r0& 4x& +10& \\ \text{multicolumn}2r& 4x& +6& \\ \text{cline}3-4\text{multicolumn}2r& 0& 4& \end{array}$$Hence, we can write $$\frac{2x^2+7x+10}{2x+3}=x+2+\frac{4}{2x+3}.$$So we can see that as $x$ becomes far from $0$, the graph of the function gets closer and closer to the line $\boxed{y=x+2}.$