Let  $f:R\to (-\infty ,-1]$ be a function defined by  $f\left(x\right)=(ab+2a-b-2)x^5-(a^3-2a+1)x^3+(a^2-2a-3)x^2+(a^2+2b)x-5,$ $a,b\in R$. If  $f\left(x\right)$ is surjective then the possible value of a-b is _______

$$\begin{gathered} f:R\to (-\infty ,-1] \\ f\left(x\right)=(ab+2a-b-2)x^5-(a^3-2a+1)x^3+(a^2-2a-3)x^2+(a^2+2b)x-5 \\ =(a-1)(b+2x)x^5-(a^3-2a+1)x^3+(a^2-2a-3)x^2+(a^2+2b)x-5 \end{gathered}$$

If f(x) is a polynomial of odd degree then its range will be R.
Thus, $a=1$ 
$$\begin{gathered} f\left(x\right)=(1-2-3)x^2+(1+2b)x-5 \\ =-4x^2+(1+2b)x-5 \end{gathered}$$

Max. Value of $f\left(x\right) = -1$

$$\begin{gathered} -\frac{D}{4a}=-1;-\frac{{(1+2b)}^2-4(-4)(-5)}{4(-4)}=-1 \\ {(1+2b)}^2-80=-16 \\ {(1+2b)}^2=64 \\ 1+2b=\pm 8 \\ 2b = 7; 2b = -9 \\ b=\frac{7}{2}or-\frac{9}{2} \end{gathered}$$