1. The equations of the vertical asymptote $\displaystyle x = {\text{const}}$ of $\displaystyle y = \frac{{\left( {x - 1} \right)^2 }}{{2x + 3}}$:
$\displaystyle 2x + 3 = 0{\text{ }} \Leftrightarrow {\text{ }}2x = - 3{\text{ }} \Leftrightarrow {\text{ }}x = - \frac{3}{2}$ is the vertical asymptote.
2. The equations of the oblique asymptote $\displaystyle y = ax + b$ of $\displaystyle y = \frac{{\left( {x - 1} \right)^2 }}{{2x + 3}}$:
$\displaystyle a = \mathop {\lim }\limits_{x \to \infty } \frac{{f\left( x \right)}}
{x} = \mathop {\lim }\limits_{x \to \infty } \frac{{\left( {x - 1} \right)^2 }}
{{\left( {2x + 3} \right)x}} = \mathop {\lim }\limits_{x \to \infty } \frac{{x^2 - 2x + 1}}{{2x^2 + 3x}} = \mathop {\lim }\limits_{x \to \infty } \frac{{1 - \frac{2}{x} + \frac{1}{{x^2 }}}}{{2 + \frac{3}{x}}} = \frac{1}{2}.$
$\displaystyle b = \mathop {\lim }\limits_{x \to \infty } \left[ {f\left( x \right) - ax} \right] = \mathop {\lim }\limits_{x \to \infty } \left[ {\frac{{\left( {x - 1} \right)^2 }}{{2x + 3}} - \frac{x}{2}} \right] = \mathop {\lim }\limits_{x \to \infty } \frac{{ - 7x + 2}}{{4x + 6}} = \mathop {\lim }\limits_{x \to \infty } \frac{{ - 7 + \frac{2}{x}}}{{4 + \frac{6}{x}}} = \frac{{ - 7}}{4}.$
The oblique asymptote is $\displaystyle y = \frac{x}{2} - \frac{7}{4}.$