Given that
\[\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = -7,\]find
\[\begin{vmatrix} a & b & c \\ 2d + 5g & 2e + 5h & 2f + 5i \\ -g & -h & -i \end{vmatrix}.\]
We are told that
\[\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = -7.\]If we multiply the second row by 2, then we get
\[\begin{vmatrix} a & b & c \\ 2d & 2e & 2f \\ g & h & i \end{vmatrix} = -14.\]Adding five times the third row to the second row does not change the value of the determinant:
\[\begin{vmatrix} a & b & c \\ 2d + 5g & 2e + 5h & 2f + 5i \\ g & h & i \end{vmatrix} = -14.\]Then multiplying the third row by $-1$ gives us
\[\begin{vmatrix} a & b & c \\ 2d + 5g & 2e + 5h & 2f + 5i \\ -g & -h & -i \end{vmatrix} = \boxed{14}.\]