What must be added to `5x^(3)-2x^(2)+6x+7` to make the sum `x^(3)+3x^(2)-x+1`?

Options

A. `4x^(3)-5x^(2)+7x+6`
B. `-4x^(3)+5x^(2)-7x-6`
C. `4x^(3)+5x^(2)-7x+6`
D. none of these

Answer

To solve the problem, we need to find what must be added to the polynomial \(5x^3 - 2x^2 + 6x + 7\) to obtain the polynomial \(x^3 + 3x^2 - x + 1\). We can represent this mathematically as:

\[ 5x^3 - 2x^2 + 6x + 7 + A = x^3 + 3x^2 - x + 1 \]

Where \(A\) is the polynomial we need to find. To isolate \(A\), we can rearrange the equation:

\[ A = (x^3 + 3x^2 - x + 1) - (5x^3 - 2x^2 + 6x + 7) \]

Now, let's simplify the right-hand side step by step.

### Step 1: Distribute the negative sign
We will distribute the negative sign across the polynomial \(5x^3 - 2x^2 + 6x + 7\):

\[ A = x^3 + 3x^2 - x + 1 - 5x^3 + 2x^2 - 6x - 7 \]

### Step 2: Combine like terms
Now, we will combine the like terms:

1. For \(x^3\):
\[ x^3 - 5x^3 = -4x^3 \]

2. For \(x^2\):
\[ 3x^2 + 2x^2 = 5x^2 \]

3. For \(x\):
\[ -x - 6x = -7x \]

4. For the constant terms:
\[ 1 - 7 = -6 \]

Putting it all together, we have:

\[ A = -4x^3 + 5x^2 - 7x - 6 \]

### Final Answer
Thus, the polynomial that must be added is:
\[ \boxed{-4x^3 + 5x^2 - 7x - 6} \]