Value of `((x^(4)+x^(2)+1))/((x^(2)-x+1))-(x+1)^(2)` is

Options

A. x
B. `-x`
C. 2x
D. 3x

Answer

To solve the expression \(\frac{x^4 + x^2 + 1}{x^2 - x + 1} - (x + 1)^2\), we will follow these steps:

### Step 1: Simplify the numerator
The numerator is \(x^4 + x^2 + 1\). We can rewrite \(x^4\) as \((x^2)^2\) to see if we can factor it.

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

### Step 2: Factor the numerator
We can recognize that \(x^4 + x^2 + 1\) can be factored using the identity for the sum of cubes. It can be expressed as:

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

### Step 3: Rewrite the expression
Now we can rewrite the original expression:

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

Substituting the factored form of the numerator:

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

### Step 4: Simplify the fraction
The expression simplifies to:

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

### Step 5: Expand \((x + 1)^2\)
Now, we need to expand \((x + 1)^2\):

\[ (x + 1)^2 = x^2 + 2x + 1 \]

### Step 6: Combine the terms
Now we can combine the terms:

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

### Step 7: Combine like terms
Now we will combine the fractions and simplify:

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

### Step 8: Final simplification
After simplifying the numerator, we will find that:

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

### Conclusion
Thus, the value of the original expression simplifies to:

\[ -x \]