To find the partial fraction decomposition of the rational function:

$\frac{x^4 + x^3 + x^2 + 1}{x^2 + x - 2}$

Step 1: Factor the denominator

We start by factoring the denominator $x^2 + x - 2$. To do this, find the factors of $-2$ that sum up to $1$ (the coefficient of $x$):

$x^2 + x - 2 = (x - 1)(x + 2)$

Step 2: Express the function as a sum of partial fractions

Now we express the given rational function as a sum of partial fractions with the factored denominator:

$\frac{x^4 + x^3 + x^2 + 1}{(x - 1)(x + 2)} = \frac{A}{x - 1} + \frac{B}{x + 2}$

Step 3: Clear the denominator

Multiply both sides of the equation by the denominator $(x - 1)(x + 2)$ to get rid of the denominator:

$x^4 + x^3 + x^2 + 1 = A(x + 2) + B(x - 1)$

Step 4: Expand and simplify

Now expand the right-hand side:

$x^4 + x^3 + x^2 + 1 = A(x + 2) + B(x - 1)$ $= A(x) + A(2) + B(x) - B$ $= (A + B)x + (2A - B)$

Step 5: Compare coefficients

Now, compare the coefficients of powers of $x$ on both sides of the equation. For this expression, we match the coefficients of like powers of $x$ on both sides.