If `(x^(2)+3)/((x^(2)+2)(x^(2)+5))=(Ax+B)/(x^(2)+2)+(Cx+D)/(x^(2)+5)`, then find the value of `A+C`.

Answer

To solve the equation

\[ \frac{x^2 + 3}{(x^2 + 2)(x^2 + 5)} = \frac{Ax + B}{x^2 + 2} + \frac{Cx + D}{x^2 + 5} \]

we will follow these steps:

### Step 1: Clear the denominators
Multiply both sides by \((x^2 + 2)(x^2 + 5)\):

\[ x^2 + 3 = (Ax + B)(x^2 + 5) + (Cx + D)(x^2 + 2) \]

### Step 2: Expand the right-hand side
Expanding both terms on the right-hand side:

\[ (Ax + B)(x^2 + 5) = Ax^3 + 5Ax + Bx^2 + 5B \]

\[ (Cx + D)(x^2 + 2) = Cx^3 + 2Cx + Dx^2 + 2D \]

Combining these, we get:

\[ x^2 + 3 = (A + C)x^3 + (B + D)x^2 + (5A + 2C)x + (5B + 2D) \]

### Step 3: Set up equations based on coefficients
Now, we can equate the coefficients from both sides of the equation:

1. Coefficient of \(x^3\): \(A + C = 0\)
2. Coefficient of \(x^2\): \(B + D = 1\)
3. Coefficient of \(x\): \(5A + 2C = 0\)
4. Constant term: \(5B + 2D = 3\)

### Step 4: Solve for \(A + C\)
From the first equation, we already have:

\[ A + C = 0 \]

Thus, the value of \(A + C\) is \(0\).

### Final Answer
The value of \(A + C\) is \(0\).

---