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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:

\[
(5) - (2) = A + B - (A + B)
\]
\[
0 = 0
\]



Given \( \frac{2 + 5x - x^2}{x^2(x + 1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x + 1} \), find values for A, B, and C.

Learn how to determine the constants A, B, and C in the partial fraction decomposition of \( \frac{2 + 5x - x^2}{x^2(x + 1)} \) by solving a system of equations. This problem demonstrates techniques in algebra and is ideal for high school students in Grades 10-12.

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