What value must be subtracted from the polynomial `4y^4 + 12y^3 + 6y^2 + 50y + 26`, so that the resultant polynomial is exactly divisible by `y^2 + 4y + 2 ?`

Options

A. y
B. `4y+5`
C. `2y-2`
D. `2y+3`

Answer

To find the value that must be subtracted from the polynomial \(4y^4 + 12y^3 + 6y^2 + 50y + 26\) so that the resultant polynomial is exactly divisible by \(y^2 + 4y + 2\), we can follow these steps:

### Step 1: Identify the Polynomial and the Divisor
We have the polynomial:
\[ P(y) = 4y^4 + 12y^3 + 6y^2 + 50y + 26 \]
and we want to divide it by:
\[ D(y) = y^2 + 4y + 2 \]

### Step 2: Perform Polynomial Long Division
We will divide \(P(y)\) by \(D(y)\) using polynomial long division.

1. **Divide the leading term** of \(P(y)\) by the leading term of \(D(y)\):
\[ \frac{4y^4}{y^2} = 4y^2 \]
2. **Multiply** \(D(y)\) by \(4y^2\):
\[ 4y^2(y^2 + 4y + 2) = 4y^4 + 16y^3 + 8y^2 \]
3. **Subtract** this from \(P(y)\):
\[ (4y^4 + 12y^3 + 6y^2 + 50y + 26) - (4y^4 + 16y^3 + 8y^2) = -4y^3 - 2y^2 + 50y + 26 \]

### Step 3: Repeat the Division Process
1. **Divide** the leading term of the new polynomial by the leading term of \(D(y)\):
\[ \frac{-4y^3}{y^2} = -4y \]
2. **Multiply** \(D(y)\) by \(-4y\):
\[ -4y(y^2 + 4y + 2) = -4y^3 - 16y^2 - 8y \]
3. **Subtract** this from the current polynomial:
\[ (-4y^3 - 2y^2 + 50y + 26) - (-4y^3 - 16y^2 - 8y) = 14y^2 + 58y + 26 \]

### Step 4: Continue the Division
1. **Divide** the leading term:
\[ \frac{14y^2}{y^2} = 14 \]
2. **Multiply** \(D(y)\) by \(14\):
\[ 14(y^2 + 4y + 2) = 14y^2 + 56y + 28 \]
3. **Subtract** this from the current polynomial:
\[ (14y^2 + 58y + 26) - (14y^2 + 56y + 28) = 2y - 2 \]

### Step 5: Identify the Remainder
After performing the polynomial long division, we find that the remainder is:
\[ R(y) = 2y - 2 \]

### Step 6: Determine the Value to Subtract
To make the polynomial exactly divisible by \(D(y)\), we need to subtract the remainder \(R(y)\):
\[ \text{Value to subtract} = 2y - 2 \]

### Final Answer
Thus, the value that must be subtracted from the polynomial \(4y^4 + 12y^3 + 6y^2 + 50y + 26\) is:
\[ \boxed{2y - 2} \]