What must be subtracted from `(x^(4)+2x^(3)-2x^(2)+4x+6)` so that the result is exactly divisible by `(x^(2)+2x-3)`?

Options

A. 2x+9
B. 9x+2
C. `-x^2+5`
D. None of these

Explanation

To find what must be subtracted from the polynomial \( P(x) = x^4 + 2x^3 - 2x^2 + 4x + 6 \) so that the result is exactly divisible by \( Q(x) = x^2 + 2x - 3 \), we can follow these steps:

### Step 1: Factor \( Q(x) \)
First, we need to factor the quadratic polynomial \( Q(x) \).

\[ Q(x) = x^2 + 2x - 3 \]

To factor this, we look for two numbers that multiply to \(-3\) (the constant term) and add to \(2\) (the coefficient of \(x\)). The numbers \(3\) and \(-1\) work.

Thus, we can factor \( Q(x) \) as:

\[ Q(x) = (x + 3)(x - 1) \]

### Step 2: Perform Polynomial Long Division
Next, we need to perform polynomial long division of \( P(x) \) by \( Q(x) \) to find the remainder.

1. Divide the leading term of \( P(x) \) by the leading term of \( Q(x) \):
\[ \frac{x^4}{x^2} = x^2 \]

2. Multiply \( Q(x) \) by \( x^2 \):
\[ x^2(x^2 + 2x - 3) = x^4 + 2x^3 - 3x^2 \]

3. Subtract this from \( P(x) \):
\[ (x^4 + 2x^3 - 2x^2 + 4x + 6) - (x^4 + 2x^3 - 3x^2) = (1x^2 + 4x + 6) \]

### Step 3: Divide Again
Now, we need to divide the new polynomial \( x^2 + 4x + 6 \) by \( Q(x) \).

1. Divide the leading term:
\[ \frac{x^2}{x^2} = 1 \]

2. Multiply \( Q(x) \) by \( 1 \):
\[ 1(x^2 + 2x - 3) = x^2 + 2x - 3 \]

3. Subtract this from \( x^2 + 4x + 6 \):
\[ (x^2 + 4x + 6) - (x^2 + 2x - 3) = 2x + 9 \]

### Step 4: Conclusion
The remainder is \( 2x + 9 \). For \( P(x) \) to be divisible by \( Q(x) \), we need to subtract this remainder from \( P(x) \).

Thus, the expression that must be subtracted from \( P(x) \) is:

\[ \boxed{2x + 9} \]