What must be subtracted from `2x^(2)-1` in order to get `x^(3)+x^(2)+x+1`?

Answer

To find what must be subtracted from \(2x^2 - 1\) in order to obtain \(x^3 + x^2 + x + 1\), we can follow these steps:

### Step 1: Set up the equation
We need to find a polynomial \(R(x)\) such that:
\[ 2x^2 - 1 - R(x) = x^3 + x^2 + x + 1 \]

### Step 2: Rearrange the equation
We can rearrange this equation to isolate \(R(x)\):
\[ R(x) = 2x^2 - 1 - (x^3 + x^2 + x + 1) \]

### Step 3: Distribute the negative sign
Now we will distribute the negative sign across the polynomial on the right:
\[ R(x) = 2x^2 - 1 - x^3 - x^2 - x - 1 \]

### Step 4: Combine like terms
Next, we combine like terms:
- The \(x^3\) term: \(-x^3\)
- The \(x^2\) terms: \(2x^2 - x^2 = x^2\)
- The \(x\) term: \(-x\)
- The constant terms: \(-1 - 1 = -2\)

So, we have:
\[ R(x) = -x^3 + x^2 - x - 2 \]

### Step 5: Final answer
Thus, the polynomial that must be subtracted from \(2x^2 - 1\) is:
\[ R(x) = -x^3 + x^2 - x - 2 \]