There is no inverse function, because for two different values of x we get the same value for f(x).
When f(x)=(2x-3)(x+1)=0, x=1.5 or -1. The inverse function would need to map 0 onto both 1.5 and -1. But a function can only map the domain onto one unique value. The function can by definition only map 0 to 1.5 or -1, not both. Therefore the inverse would be incomplete, or there would be two possible inverses. There is no single function that could be an inverse.