Official solution
Given, (gof) (x) = 4x2 - 10x + 5
\[ \Rightarrow \] g(f(x)) = 4x2 - 10x + 5
\[ \therefore \] g(f(\[{5 \over 4}\])) = \[4 \times {{25} \over {16}} - {{50} \over 4} + 5\] = \[ - {5 \over 4}\] ...(1)
Also given, g(x) = x2 + x - 1
\[ \therefore \] g(f(x)) = f2(x) + f(x) --1
\[ \Rightarrow \] g(f(\[{5 \over 4}\])) = f2(\[{5 \over 4}\]) + f(\[{5 \over 4}\]) --1 ....(2)
from (1) \& (2)
f2(\[{5 \over 4}\]) + f(\[{5 \over 4}\]) --1 = \[ - {5 \over 4}\]
\[ \Rightarrow \] f2(\[{5 \over 4}\]) + f(\[{5 \over 4}\]) + \[{1 \over 4}\] = 0
\[ \Rightarrow \] (f(\[{5 \over 4}\]) + \[{1 \over 2}\])2 = 0
\[ \Rightarrow \] f(\[{5 \over 4}\]) = -\[{1 \over 2}\]