Resolve into partial fractions :
`(x^(2)+7x+11)/((x+2)^(2)(x+3))= (A)/((x+2))+(B)/((x+2)^2)+(C )/(x+3)`.

Answer

To resolve the expression \(\frac{x^2 + 7x + 11}{(x + 2)^2 (x + 3)}\) into partial fractions, we start by expressing it in the form:

\[ \frac{x^2 + 7x + 11}{(x + 2)^2 (x + 3)} = \frac{A}{(x + 2)} + \frac{B}{(x + 2)^2} + \frac{C}{(x + 3)} \]

### Step 1: Set up the equation
We multiply both sides by the denominator \((x + 2)^2 (x + 3)\) to eliminate the fractions:

\[ x^2 + 7x + 11 = A(x + 2)(x + 3) + B(x + 3) + C(x + 2)^2 \]

### Step 2: Expand the right-hand side
Now we expand the right-hand side:

1. For \(A(x + 2)(x + 3)\):
\[ A(x^2 + 5x + 6) = Ax^2 + 5Ax + 6A \]

2. For \(B(x + 3)\):
\[ Bx + 3B \]

3. For \(C(x + 2)^2\):
\[ C(x^2 + 4x + 4) = Cx^2 + 4Cx + 4C \]

Combining these gives:

\[ x^2 + 7x + 11 = (A + C)x^2 + (5A + B + 4C)x + (6A + 3B + 4C) \]

### Step 3: Compare coefficients
Now we compare the coefficients of both sides:

1. Coefficient of \(x^2\):
\[ A + C = 1 \quad \text{(1)} \]

2. Coefficient of \(x\):
\[ 5A + B + 4C = 7 \quad \text{(2)} \]

3. Constant term:
\[ 6A + 3B + 4C = 11 \quad \text{(3)} \]

### Step 4: Solve the system of equations
We have the system of equations:

1. \(A + C = 1\)
2. \(5A + B + 4C = 7\)
3. \(6A + 3B + 4C = 11\)

From equation (1), we can express \(C\) in terms of \(A\):
\[ C = 1 - A \quad \text{(4)} \]

Substituting (4) into equations (2) and (3):

**Substituting into (2):**
\[ 5A + B + 4(1 - A) = 7 \]
\[ 5A + B + 4 - 4A = 7 \]
\[ A + B + 4 = 7 \implies B = 3 - A \quad \text{(5)} \]

**Substituting into (3):**
\[ 6A + 3(3 - A) + 4(1 - A) = 11 \]
\[ 6A + 9 - 3A + 4 - 4A = 11 \]
\[ -1A + 13 = 11 \implies -A = -2 \implies A = 2 \]

### Step 5: Find \(B\) and \(C\)
Using \(A = 2\) in (4) and (5):

From (4):
\[ C = 1 - 2 = -1 \]

From (5):
\[ B = 3 - 2 = 1 \]

### Step 6: Write the final partial fraction decomposition
Now substituting \(A\), \(B\), and \(C\) back into the partial fractions:

\[ \frac{x^2 + 7x + 11}{(x + 2)^2 (x + 3)} = \frac{2}{(x + 2)} + \frac{1}{(x + 2)^2} - \frac{1}{(x + 3)} \]

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

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