If `(x^(3)+x^(2)+1)/((x^(2)+2)(x^(2)+3))=(Ax+B)/(x^(2)+2)+(Cx+D)/(x^(2)+3)`, then `A+B+C+D=`

Options

A. 1
B. 4
C. 3
D. 2

Answer

To solve the equation

\[ \frac{x^3 + x^2 + 1}{(x^2 + 2)(x^2 + 3)} = \frac{Ax + B}{x^2 + 2} + \frac{Cx + D}{x^2 + 3}, \]

we will follow these steps:

### Step 1: Combine the Right Side
First, we will combine the right-hand side into a single fraction:

\[ \frac{Ax + B}{x^2 + 2} + \frac{Cx + D}{x^2 + 3} = \frac{(Ax + B)(x^2 + 3) + (Cx + D)(x^2 + 2)}{(x^2 + 2)(x^2 + 3)}. \]

### Step 2: Expand the Numerator
Now, we will expand the numerator:

\[ (Ax + B)(x^2 + 3) = Ax^3 + 3Ax + Bx^2 + 3B, \]
\[ (Cx + D)(x^2 + 2) = Cx^3 + 2Cx + Dx^2 + 2D. \]

Combining these, we get:

\[ Ax^3 + 3Ax + Bx^2 + 3B + Cx^3 + 2Cx + Dx^2 + 2D = (A + C)x^3 + (B + D)x^2 + (3A + 2C)x + (3B + 2D). \]

### Step 3: Set the Numerators Equal
Now we set the numerators equal to each other:

\[ x^3 + x^2 + 1 = (A + C)x^3 + (B + D)x^2 + (3A + 2C)x + (3B + 2D). \]

### Step 4: Compare Coefficients
Now we will compare the coefficients of both sides:

1. Coefficient of \(x^3\):
\[ 1 = A + C \quad \text{(Equation 1)} \]

2. Coefficient of \(x^2\):
\[ 1 = B + D \quad \text{(Equation 2)} \]

3. Coefficient of \(x\):
\[ 0 = 3A + 2C \quad \text{(Equation 3)} \]

4. Constant term:
\[ 1 = 3B + 2D \quad \text{(Equation 4)} \]

### Step 5: Solve the Equations
From Equation 1, we have:

\[ C = 1 - A. \]

Substituting \(C\) into Equation 3:

\[ 0 = 3A + 2(1 - A) \implies 0 = 3A + 2 - 2A \implies A = -2. \]

Now substituting \(A\) back into Equation 1:

\[ 1 = -2 + C \implies C = 3. \]

Now substituting \(A\) into Equation 2:

\[ 1 = B + D \quad \text{(Equation 2)} \]

Substituting \(B\) into Equation 4:

\[ 1 = 3B + 2D. \]

We can express \(D\) from Equation 2:

\[ D = 1 - B. \]

Substituting \(D\) into Equation 4:

\[ 1 = 3B + 2(1 - B) \implies 1 = 3B + 2 - 2B \implies B = -1. \]

Now substituting \(B\) back into Equation 2:

\[ 1 = -1 + D \implies D = 2. \]

### Step 6: Calculate \(A + B + C + D\)
Now we have:

- \(A = -2\)
- \(B = -1\)
- \(C = 3\)
- \(D = 2\)

Calculating \(A + B + C + D\):

\[ A + B + C + D = -2 - 1 + 3 + 2 = 2. \]

### Final Answer
Thus, the value of \(A + B + C + D\) is

\[ \boxed{2}. \]