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

Answer

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

### Step 1: Set Up the Partial Fraction Decomposition
We express the given fraction as a sum of simpler fractions. Since the denominator is a square of a quadratic polynomial, we can write:
\[ \frac{x^2 + 4x + 7}{(x^2 + x + 1)^2} = \frac{Ax + B}{x^2 + x + 1} + \frac{Cx + D}{(x^2 + x + 1)^2} \]
where \(A\), \(B\), \(C\), and \(D\) are constants we need to determine.

### Step 2: Clear the Denominator
Multiply both sides by \((x^2 + x + 1)^2\) to eliminate the denominator:
\[ x^2 + 4x + 7 = (Ax + B)(x^2 + x + 1) + (Cx + D) \]

### Step 3: Expand the Right Side
Now, expand the right-hand side:
\[ (Ax + B)(x^2 + x + 1) = Ax^3 + Ax^2 + Ax + Bx^2 + Bx + B \]
Combining like terms gives:
\[ Ax^3 + (A + B)x^2 + (A + B)x + B \]
Adding \(Cx + D\) gives:
\[ Ax^3 + (A + B)x^2 + (A + B + C)x + (B + D) \]

### Step 4: Set Up the Equation
Now we equate the coefficients from both sides:
\[ x^2 + 4x + 7 = Ax^3 + (A + B)x^2 + (A + B + C)x + (B + D) \]

### Step 5: Compare Coefficients
From the left side, we have:
- Coefficient of \(x^3\): \(0 = A\)
- Coefficient of \(x^2\): \(1 = A + B\)
- Coefficient of \(x\): \(4 = A + B + C\)
- Constant term: \(7 = B + D\)

### Step 6: Solve the System of Equations
1. From \(0 = A\), we have \(A = 0\).
2. Substitute \(A = 0\) into \(1 = A + B\):
\[ 1 = 0 + B \implies B = 1 \]
3. Substitute \(A = 0\) and \(B = 1\) into \(4 = A + B + C\):
\[ 4 = 0 + 1 + C \implies C = 3 \]
4. Substitute \(B = 1\) into \(7 = B + D\):
\[ 7 = 1 + D \implies D = 6 \]

### Step 7: Write the Final Partial Fraction Decomposition
Now we can substitute \(A\), \(B\), \(C\), and \(D\) back into the partial fractions:
\[ \frac{x^2 + 4x + 7}{(x^2 + x + 1)^2} = \frac{0 \cdot x + 1}{x^2 + x + 1} + \frac{3x + 6}{(x^2 + x + 1)^2} \]
This simplifies to:
\[ \frac{1}{x^2 + x + 1} + \frac{3x + 6}{(x^2 + x + 1)^2} \]

### Final Answer
Thus, the partial fraction decomposition is:
\[ \frac{1}{x^2 + x + 1} + \frac{3x + 6}{(x^2 + x + 1)^2} \]