The given equation is:

$\frac{2 + 5x - x^2}{x^2(x + 1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x + 1}$

We are tasked with determining the values of $A$, $B$, and $C$ by performing partial fraction decomposition.

Step 1: Express the equation in terms of a single denominator.

The right-hand side of the equation needs to be combined into a single fraction with the denominator $x^2(x + 1)$. Start by adding the fractions:

$\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1}$

The least common denominator is $x^2(x + 1)$. Rewrite each term with this common denominator:

$\frac{A}{x} = \frac{A(x + 1)}{x^2(x + 1)}, \quad \frac{B}{x^2} = \frac{B(x + 1)}{x^2(x + 1)}, \quad \frac{C}{x + 1} = \frac{C x^2}{x^2(x + 1)}$

Thus, the right-hand side becomes:

$\frac{A(x + 1) + B(x + 1) + Cx^2}{x^2(x + 1)}$

Step 2: Set up the equation.

Now, set the numerators equal to each other:

$2 + 5x - x^2 = A(x + 1) + B(x + 1) + Cx^2$

Step 3: Expand and collect like terms.

Expand both sides:

$2 + 5x - x^2 = A(x + 1) + B(x + 1) + Cx^2$

$2 + 5x - x^2 = (A + B)(x + 1) + Cx^2$

$2 + 5x - x^2 = (A + B)x + (A + B) + Cx^2$

Now group terms:

$2 + 5x - x^2 = Cx^2 + (A + B)x + (A + B)$

Step 4: Equate the coefficients.

By comparing the coefficients of $x^2$, $x$, and the constant terms, we get the following system of equations:

1. Coefficient of $x^2$: $-1 = C$
2. Coefficient of $x$: $5 = A + B$
3. Constant term: $2 = A + B$

Step 5: Solve the system of equations.

From equation (1), we know $C = -1$.

Now substitute $A + B = 5$ into the constant equation $A + B = 2$. We can subtract these two equations:

$(5) - (2) = A + B - (A + B)$ $0 = 0$