Answer
$x=-\frac{5}{8}\pm\frac{3\sqrt{17}}{8}$
Work Step by Step
We complete the square:
$4x^{2}+5x-8=0$
$x^{2}+\frac{5}{4}x-2=0$
$x^{2}+\frac{5}{4}x=2$
$x^{2}+\frac{5}{4}x+\frac{25}{64}=2+\frac{25}{64}=\frac{153}{64}$
$(x+\frac{5}{8})^{2}=\frac{153}{64}$
$x+\frac{5}{8}=\pm\sqrt{\frac{153}{64}}=\pm\frac{3\sqrt{17}}{8}$
$x=-\frac{5}{8}\pm\frac{3\sqrt{17}}{8}$