\(\displaystyle f(x)\) is quadratic ...

\(\displaystyle f(x) = ax^2 + bx + c\)

\(\displaystyle f(2x-3) = a(2x-3)^2 + b(2x-3) + c = x^2 + x - 1\)

\(\displaystyle f(2x-3) = a(4x^2-12x+9) + b(2x-3) + c = x^2 + x - 1\)

\(\displaystyle 4ax^2 + (2b-12a)x + (9a-3b+c) = x^2 + x - 1\)

equate coefficients ...

\(\displaystyle 4a = 1\) ... \(\displaystyle a = \frac{1}{4}\)

\(\displaystyle 2b - 3 = 1\) ... \(\displaystyle b = 2\)

\(\displaystyle \frac{9}{4} - 6 + c = -1\) ... \(\displaystyle c = \frac{11}{4}\)

\(\displaystyle f(x) = \frac{x^2}{4} + 2x + \frac{11}{4}\)