Resolve `(2x+7)/((x+1)(x^(2)+4))` into partial fractions.

Answer

To resolve the expression \(\frac{2x + 7}{(x + 1)(x^2 + 4)}\) into partial fractions, we will follow these steps:

### Step 1: Set Up the Partial Fraction Decomposition
We start by expressing the given fraction in terms of partial fractions. The denominator consists of a linear factor \((x + 1)\) and an irreducible quadratic factor \((x^2 + 4)\). Therefore, we can write:

\[ \frac{2x + 7}{(x + 1)(x^2 + 4)} = \frac{A}{x + 1} + \frac{Bx + C}{x^2 + 4} \]

where \(A\), \(B\), and \(C\) are constants that we need to determine.

### Step 2: Clear the Denominator
Next, we multiply both sides of the equation by the denominator \((x + 1)(x^2 + 4)\) to eliminate the fractions:

\[ 2x + 7 = A(x^2 + 4) + (Bx + C)(x + 1) \]

### Step 3: Expand the Right Side
Now, we expand the right-hand side:

\[ A(x^2 + 4) + (Bx + C)(x + 1) = Ax^2 + 4A + Bx^2 + Bx + Cx + C \]

Combining like terms, we get:

\[ (A + B)x^2 + (B + C)x + (4A + C) \]

### Step 4: Equate Coefficients
Now, we equate the coefficients from both sides of the equation:

1. Coefficient of \(x^2\): \(A + B = 0\)
2. Coefficient of \(x\): \(B + C = 2\)
3. Constant term: \(4A + C = 7\)

### Step 5: Solve the System of Equations
We now have a system of equations to solve:

1. From \(A + B = 0\), we can express \(B\) in terms of \(A\):
\[ B = -A \]

2. Substitute \(B = -A\) into the second equation \(B + C = 2\):
\[ -A + C = 2 \implies C = A + 2 \]

3. Substitute \(B = -A\) and \(C = A + 2\) into the third equation \(4A + C = 7\):
\[ 4A + (A + 2) = 7 \implies 5A + 2 = 7 \implies 5A = 5 \implies A = 1 \]

4. Now substitute \(A = 1\) back to find \(B\) and \(C\):
\[ B = -1 \quad \text{and} \quad C = 1 + 2 = 3 \]

### Step 6: Write the Partial Fraction Decomposition
Now that we have found \(A\), \(B\), and \(C\), we can write the partial fraction decomposition:

\[ \frac{2x + 7}{(x + 1)(x^2 + 4)} = \frac{1}{x + 1} + \frac{-x + 3}{x^2 + 4} \]

### Final Answer
Thus, the partial fraction decomposition is:

\[ \frac{2x + 7}{(x + 1)(x^2 + 4)} = \frac{1}{x + 1} + \frac{-x + 3}{x^2 + 4} \]