What must be added to `x^(4) - x^(3) + x^(2) + x + 3` to obtain `x^(4) + x^(2) - 1` ?

Answer

To find what must be added to the expression \( x^{4} - x^{3} + x^{2} + x + 3 \) to obtain \( x^{4} + x^{2} - 1 \), we need to perform the following steps:

### Step 1: Set up the equation
We want to find a value \( A \) such that:
\[ (x^{4} - x^{3} + x^{2} + x + 3) + A = x^{4} + x^{2} - 1 \]
To find \( A \), we can rearrange this equation:
\[ A = (x^{4} + x^{2} - 1) - (x^{4} - x^{3} + x^{2} + x + 3) \]

### Step 2: Substitute the expressions
Now, we will substitute the expressions into the equation:
\[ A = (x^{4} + x^{2} - 1) - (x^{4} - x^{3} + x^{2} + x + 3) \]

### Step 3: Distribute the negative sign
Distributing the negative sign across the second expression gives:
\[ A = x^{4} + x^{2} - 1 - x^{4} + x^{3} - x^{2} - x - 3 \]

### Step 4: Combine like terms
Now, we can combine like terms:
- The \( x^{4} \) terms: \( x^{4} - x^{4} = 0 \)
- The \( x^{3} \) term: \( 0 + x^{3} = x^{3} \)
- The \( x^{2} \) terms: \( x^{2} - x^{2} = 0 \)
- The \( x \) term: \( 0 - x = -x \)
- The constant terms: \( -1 - 3 = -4 \)

Putting it all together, we have:
\[ A = x^{3} - x - 4 \]

### Final Answer
Thus, the expression that must be added is:
\[ \boxed{x^{3} - x - 4} \]