jmc

Denote the roots of $5x^2+3x+4$ by $a$ and $b$. We have that $\alpha =\frac{1}{a}$ and $\beta =\frac{1}{b}$. So, $$\alpha +\beta =\frac{1}{a}+\frac{1}{b}=\frac{a+b}{ab}.$$

Now, we know that $a+b=\frac{-3}{5}$ and $ab=\frac{4}{5}$ by the relationship between sum/products of roots and the coefficients of a polynomial.

Hence $\alpha +\beta =\frac{a+b}{ab}=\boxed{-\frac{3}{4}}$.