What polynomial must be added ot `7x^2 + 14x - 8` to result in a sum of `5x^2 + 18x + 1` ?

Options

A. `-2x^2- 4x +7`
B. `2x^2-4x +9`
C. `-2x^2 + 4x + 7`
D. `2x + 4x + 9`

Answer

To find the polynomial that must be added to \(7x^2 + 14x - 8\) to achieve a sum of \(5x^2 + 18x + 1\), we can set up the equation as follows:

1. **Set up the equation**: Let the unknown polynomial be \(P(x)\). We can write the equation:
\[ 7x^2 + 14x - 8 + P(x) = 5x^2 + 18x + 1 \]

2. **Rearrange the equation**: To isolate \(P(x)\), we can rearrange the equation:
\[ P(x) = (5x^2 + 18x + 1) - (7x^2 + 14x - 8) \]

3. **Distribute the negative sign**: When we subtract the polynomial \(7x^2 + 14x - 8\), we need to distribute the negative sign:
\[ P(x) = 5x^2 + 18x + 1 - 7x^2 - 14x + 8 \]

4. **Combine like terms**: Now we combine the like terms:
- For \(x^2\) terms: \(5x^2 - 7x^2 = -2x^2\)
- For \(x\) terms: \(18x - 14x = 4x\)
- For constant terms: \(1 + 8 = 9\)

Putting it all together, we have:
\[ P(x) = -2x^2 + 4x + 9 \]

5. **Final answer**: Therefore, the polynomial that must be added is:
\[ P(x) = -2x^2 + 4x + 9 \]