What polynomial must be added to `x^(2)+3x-5` so that the sum is `5x^(2)-8`?

Options

A. `4x^(2)-5x+6`
B. `4x^(2)-3x-3`
C. `5x^(2)-3x-3`
D. `5x^(2)+3x+6`

Answer

To find the polynomial that must be added to \(x^2 + 3x - 5\) so that the sum equals \(5x^2 - 8\), we can follow these steps:

### Step 1: Define the polynomial to be added
Let the polynomial that we need to find be denoted as \(y\).

### Step 2: Set up the equation
According to the problem, we have:
\[ x^2 + 3x - 5 + y = 5x^2 - 8 \]

### Step 3: Isolate \(y\)
To find \(y\), we can rearrange the equation:
\[ y = (5x^2 - 8) - (x^2 + 3x - 5) \]

### Step 4: Distribute the negative sign
Now, we will distribute the negative sign across the terms in the parentheses:
\[ y = 5x^2 - 8 - x^2 - 3x + 5 \]

### Step 5: Combine like terms
Next, we will combine the like terms:
- For \(x^2\): \(5x^2 - x^2 = 4x^2\)
- For \(x\): \(-3x\) (there's no other \(x\) term to combine with)
- For the constant terms: \(-8 + 5 = -3\)

Putting it all together, we have:
\[ y = 4x^2 - 3x - 3 \]

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

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