Prove function is continuous using epsilon delta

Hello,

The function given is f(z)=(z^2+3)/(z+1) and the domain is the complex plane C\{-1}.
Now let epsilon e>0 be given, so that
|f(z)-f(zo)|=|(z-zo)(z^2-zo^2+zzo-3)/(…
now I can set delta d=(z+1)(zo+1)e/(2(z^2+zo^2+zzo-3)) and it will work but I can't show that
the denominator of delta is not zero..or even factor.
Any help will be appreciated.

Thank you

Note that |f(z) - f(z₀)|
= |[(z^2 + 3)(z₀ + 1) - (z₀^2 + 3)(z + 1)] / [(z₀ + 1)(z + 1)]|
= |[(z₀ z^2 + z^2 + 3z₀ + 3) - (z z₀^2 + z₀^2 + 3z + 3)] / [(z₀ + 1)(z + 1)]|.

Expand the numerator into powers of (z - z₀):
|f(z) - f(z₀)|
= |[(z₀ z^2 - z z₀^2) + (z^2 - z�