Inverse Z-transform with complex conjugate poles
Since the given $\mathcal{Z}$-transform

$$X(z)=\frac{2+3z^{-1}}{1-z^{-1}+0.81z^{-2}},\quad |z|&gt;0.9\tag{1}$$

has real-valued coefficients, its poles and zeros must be either real-valued or they must occur in complex conjugate pairs. The poles of $(1)$ are a complex conjugate pair:

$$p_1=0.5 + 0.74833j=re^{j\phi}=p\\
p_2=0.5 - 0.74833j=re^{-j\phi}=p^*$$

with $r=0.9$ and $\phi=0.98177$.

You can proceed from the partial fraction expansion of $X(z)$, which you correctly obtained:

$$X(z)=\frac{A}{1-pz^{-1}}+\frac{A^*}{1-p^*z^{-1}}$$

With the given region of convergence (ROC) $|z|&gt;0.9$, we know that we're looking for a causal sequence $x[n]$:

$$x[n]=\left(Ap^n+A^*(p^*)^n\right) u[n]\tag{3}$$

With $p=re^{j\phi}$ and $A=|A|e^{j\phi_A}$, $(3)$ can be rewritten as

$$x[n]=|A|r^n\left(e^{j(n\phi+\phi_A)}+e^{-j(n\phi+\phi_A)}\right)u[n]=
2|A|r^n\cos(n\phi+\phi_A)u[n]\tag{4}$$

which is of course real-valued.

With the given values of $p$ and $A$ the impulse response looks like this: