Resolve the following into partial fractions. `(x+4)/((x^(2)-4)(x+1))`

Answer

To resolve the expression \(\frac{x+4}{(x^2-4)(x+1)}\) into partial fractions, we will follow these steps:

### Step 1: Factor the Denominator
The denominator \(x^2 - 4\) can be factored using the difference of squares:
\[ x^2 - 4 = (x - 2)(x + 2) \]
Thus, the full denominator becomes:
\[ (x - 2)(x + 2)(x + 1) \]

### Step 2: Set Up the Partial Fraction Decomposition
We can express \(\frac{x + 4}{(x - 2)(x + 2)(x + 1)}\) as a sum of partial fractions:
\[ \frac{x + 4}{(x - 2)(x + 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 2} + \frac{C}{x + 1} \]
where \(A\), \(B\), and \(C\) are constants to be determined.

### Step 3: Combine the Right Side
To combine the right side into a single fraction, we find a common denominator:
\[ \frac{A}{x - 2} + \frac{B}{x + 2} + \frac{C}{x + 1} = \frac{A(x + 2)(x + 1) + B(x - 2)(x + 1) + C(x - 2)(x + 2)}{(x - 2)(x + 2)(x + 1)} \]

### Step 4: Set the Numerators Equal
Now, we equate the numerators:
\[ x + 4 = A(x + 2)(x + 1) + B(x - 2)(x + 1) + C(x - 2)(x + 2) \]

### Step 5: Expand the Right Side
Expanding the right-hand side:
1. \(A(x + 2)(x + 1) = A(x^2 + 3x + 2)\)
2. \(B(x - 2)(x + 1) = B(x^2 - x - 2)\)
3. \(C(x - 2)(x + 2) = C(x^2 - 4)\)

Combining these gives:
\[ Ax^2 + 3Ax + 2A + Bx^2 - Bx - 2B + Cx^2 - 4C \]
Grouping by powers of \(x\):
\[ (A + B + C)x^2 + (3A - B)x + (2A - 2B - 4C) \]

### Step 6: Set Up the System of Equations
Now, we can set up equations by comparing coefficients:
1. \(A + B + C = 0\) (coefficient of \(x^2\))
2. \(3A - B = 1\) (coefficient of \(x\))
3. \(2A - 2B - 4C = 4\) (constant term)

### Step 7: Solve the System of Equations
From equation 1, we can express \(C\):
\[ C = -A - B \]
Substituting \(C\) into equation 3:
\[ 2A - 2B - 4(-A - B) = 4 \implies 2A - 2B + 4A + 4B = 4 \implies 6A + 2B = 4 \implies 3A + B = 2 \quad \text{(Equation 4)} \]

Now we have:
1. \(3A - B = 1\) (Equation 2)
2. \(3A + B = 2\) (Equation 4)

Adding these two equations:
\[ (3A - B) + (3A + B) = 1 + 2 \implies 6A = 3 \implies A = \frac{1}{2} \]

Substituting \(A = \frac{1}{2}\) into Equation 4:
\[ 3\left(\frac{1}{2}\right) + B = 2 \implies \frac{3}{2} + B = 2 \implies B = 2 - \frac{3}{2} = \frac{1}{2} \]

Now substituting \(A\) and \(B\) back to find \(C\):
\[ C = -\frac{1}{2} - \frac{1}{2} = -1 \]

### Step 8: Write the Partial Fraction Decomposition
Now we can write the partial fractions:
\[ \frac{x + 4}{(x - 2)(x + 2)(x + 1)} = \frac{1/2}{x - 2} + \frac{1/2}{x + 2} - \frac{1}{x + 1} \]

### Final Answer
Thus, the partial fraction decomposition of \(\frac{x + 4}{(x^2 - 4)(x + 1)}\) is:
\[ \frac{1/2}{x - 2} + \frac{1/2}{x + 2} - \frac{1}{x + 1} \]