  In this work we shall address the ghost issue of $F\left( R,\mathcal{G}
\right)$ gravity, which is known to be plagued with ghost degrees of freedom.
These ghosts occur due to the presence of higher than two derivatives in the
field equations, and can arise even when considering cosmological
perturbations, where superluminal modes may arise in the theory. If we consider
the quantum theory, the ghosts generate the negative norm states, which give
the negative probabilities, and therefore the ghosts are physically
inconsistent. Motivated by the importance of $F\left( R,\mathcal{G} \right)$
gravity for providing viable inflationary and dark energy phenomenologies, in
this work we shall provide a technique that can render $F\left( R,\mathcal{G}
\right)$ gravity theories free from ghost degrees of freedom. This will be done
by introducing two auxiliary scalar fields, and by employing the Lagrange
multiplier technique, the theory is ghost free in the Einstein frame. Also the
framework can be viewed as a reconstruction technique and can be used as a
method in order to realize several cosmological evolutions of interest. We
demonstrate how we can realize several cosmologically interesting
phenomenologies by using the reconstruction technique.
