Remember that the parabola's vertex was

$\left(-\frac{3}{4}, -\frac{1}{8}\right)$

Remember that the only way for a parabola to have only one $x$-intercept is to have the vertex touching the $x$-axis.

To make it so the equation only has one $x$-intercept, you must make the vertex's $y$ value equal $0$.

$\textit{y}=2\textit{x}^{2}+3\textit{x}+\frac{9}{8}$