`(1)/((x^(2)+1)^(2)(x^(2)+3))=(1)/(2)*(1)/((x^(2)+1)^2)-(1)/(4)* (1)/(x^(2)+1)+ (1)/(4)*(1)/((x^(2)+3))`

Answer

To solve the equation

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

we will verify if both sides are equal by simplifying the right-hand side and comparing it to the left-hand side.

### Step 1: Find a common denominator for the right-hand side

The common denominator for the right-hand side is \((x^2 + 1)^2 (x^2 + 3)\).

### Step 2: Rewrite each term with the common denominator

1. The first term:
\[ \frac{1}{2} \cdot \frac{1}{(x^2 + 1)^2} = \frac{1}{2} \cdot \frac{(x^2 + 3)}{(x^2 + 3)(x^2 + 1)^2} = \frac{(x^2 + 3)}{2(x^2 + 1)^2 (x^2 + 3)}. \]

2. The second term:
\[ -\frac{1}{4} \cdot \frac{1}{(x^2 + 1)} = -\frac{1}{4} \cdot \frac{(x^2 + 1)(x^2 + 3)}{(x^2 + 1)(x^2 + 3)(x^2 + 1)} = -\frac{(x^2 + 1)(x^2 + 3)}{4(x^2 + 1)(x^2 + 3)(x^2 + 1)}. \]

3. The third term:
\[ \frac{1}{4} \cdot \frac{1}{(x^2 + 3)} = \frac{1}{4} \cdot \frac{(x^2 + 1)^2}{(x^2 + 1)^2 (x^2 + 3)} = \frac{(x^2 + 1)^2}{4(x^2 + 1)^2 (x^2 + 3)}. \]

### Step 3: Combine the terms

Now we can combine all three terms over the common denominator:

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

### Step 4: Simplify the right-hand side

Combine the numerators:

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

Now, simplify the numerator:

1. Expand \((x^2 + 1)(x^2 + 3)\):
\[ = x^4 + 3x^2 + x^2 + 3 = x^4 + 4x^2 + 3. \]

2. Expand \((x^2 + 1)^2\):
\[ = x^4 + 2x^2 + 1. \]

Now substitute back into the numerator:

\[ = 2(x^2 + 3) - (x^4 + 4x^2 + 3) + (x^4 + 2x^2 + 1). \]

Combine like terms:

\[ = 2x^2 + 6 - x^4 - 4x^2 - 3 + x^4 + 2x^2 + 1 = 0. \]

### Step 5: Final comparison

Thus, the right-hand side simplifies to:

\[ \frac{0}{4(x^2 + 1)^2 (x^2 + 3)} = 0. \]

Since the left-hand side is not equal to zero, we can conclude:

\[ \frac{1}{(x^2 + 1)^2 (x^2 + 3)} \neq 0. \]

### Conclusion

The original statement is false.

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