Answer
Two rational solutions.
Work Step by Step
$ 4x^{2}+8x-5=0\qquad$.... $a=4,\ b=8,\ c=-5$
$ b^{2}-4ac\qquad$....substitute $b$ for $8,\ a$ for $4$ and $c$ for $-5$
$=(8)^{2}-4\cdot 4\cdot(-5)$
$=64+80$
$=144$
Since the discriminant is a positive number that is a perfect square, there are two rational solutions.