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To decompose the given expression into partial fractions, we start with the following rational function:

\[
\frac{16 + 5x - 2x^2}{(x+1)^2 (x+4)}
\]

### Step 1: Set up the form for partial fractions
Since the denominator has the factors \((x+1)^2\) and \((x+4)\), the corresponding partial fraction decomposition will be:

\[
\frac{16 + 5x - 2x^2}{(x+1)^2 (x+4)} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{x+4}
\]

where \(A\), \(B\), and \(C\) are constants to be determined.

### Step 2: Combine the fractions on the right-hand side
To combine these fractions over a common denominator \((x+1)^2 (x+4)\), we have:

\[
\frac{A}{x+1} = \frac{A(x+1)(x+4)}{(x+1)^2(x+4)}
\]
\[
\frac{B}{(x+1)^2} = \frac{B(x+4)}{(x+1)^2(x+4)}
\]
\[
\frac{C}{x+4} = \frac{C(x+1)^2}{(x+1)^2(x+4)}
\]

Now, summing these terms, we get:

\[
\frac{A(x+4) + B(x+4) + C(x+1)^2}{(x+1)^2(x+4)}
\]

### Step 3: Set up the equation
The numerator of the original fraction \(16 + 5x - 2x^2\) must equal the numerator of the combined partial fractions:

\[
16 + 5x - 2x^2 = A(x+4) + B(x+4) + C(x+1)^2
\]

Expand both sides to find \(A\), \(B\), and \(C\).

#### Left-hand side:
\[
16 + 5x - 2x^2
\]

#### Right-hand side:
Expanding the right-hand side:
\[
A(x+4) + B(x+4) + C(x+1)^2 = A(x+4) + B(x+4) + C(x^2 + 2x + 1)
\]
\[
= (A+B)x + (4A + 4B) + C(x^2 + 2x + 1)
\]
\[
= Cx^2 + (A + B + 2C)x + (4A + 4B + C)
\]

### Step 4: Equate coefficients
Now, compare coefficients of powers of \(x\) on both sides:

- Coefficient of \(x^2\): \(-2 = C\), so \(C = -2\)
- Coefficient of \(x\): \(5 = A + B + 2C = A + B + 2(-2) = A + B - 4\), so \(A + B = 9\)
- Constant term: \(16 = 4A + 4B + C = 4A + 4B - 2\), so \(4A + 4B = 18\), or \(A + B = \frac{9}{2}\)

### Step 5: Solve the system of equations
We now solve for \(A\) and \(B\:


Decompose the expression (16 + 5x - 2x^2) / ((x + 1)^2 (x + 4)) into partial fractions.

Learn how to decompose the rational function (16 + 5x - 2x^2) / ((x+1)^2(x+4)) into partial fractions with a detailed step-by-step solution. This tutorial covers key concepts of partial fraction decomposition, useful in algebra and calculus. Suitable for high school students in Grades 10-12.

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