To find the inverse Z-transform of $X(z) = \frac{z + 1}{z^2 - 3z + 2}$ using long division, we will perform the division and identify the resulting sequence in the time domain.

Step 1: Perform Long Division

Given: $X(z) = \frac{z + 1}{z^2 - 3z + 2}$

First, let's factor the denominator if possible: $z^2 - 3z + 2 = (z - 1)(z - 2)$

Now, perform the long division:

1. Divide the highest degree term in the numerator by the highest degree term in the denominator: $\frac{z}{z^2} = \frac{1}{z}$ But we need terms in decreasing powers of $z$. This step suggests we'll rewrite it as $0$ for this stage of division and continue for lower degree terms.

Since ( \frac{z}{z^