On dividing the polynomial
`8x ^(3) - 6x (2) + 10 x + 3 by (4x +1),` the quotient is `2x ^(2) + k`, where, k is equal to

Options

A. `3 -2 x`
B. `- 3 + 2x`
C. `3 + 2x`
D. `- 3 - 2x`

Answer

To find the value of \( k \) in the quotient of the polynomial division of \( 8x^3 - 6x^2 + 10x + 3 \) by \( 4x + 1 \), we can use the long division method. Here are the steps:

### Step 1: Set up the division
We start by writing the polynomial \( 8x^3 - 6x^2 + 10x + 3 \) under the long division symbol and \( 4x + 1 \) outside.

### Step 2: Divide the leading terms
Divide the leading term of the dividend \( 8x^3 \) by the leading term of the divisor \( 4x \):
\[ \frac{8x^3}{4x} = 2x^2 \]
This is the first term of our quotient.

### Step 3: Multiply and subtract
Now, multiply \( 2x^2 \) by the entire divisor \( 4x + 1 \):
\[ 2x^2 \cdot (4x + 1) = 8x^3 + 2x^2 \]
Subtract this from the original polynomial:
\[ (8x^3 - 6x^2 + 10x + 3) - (8x^3 + 2x^2) = -6x^2 - 2x^2 + 10x + 3 = -8x^2 + 10x + 3 \]

### Step 4: Repeat the process
Now, take the new polynomial \( -8x^2 + 10x + 3 \) and divide the leading term \( -8x^2 \) by \( 4x \):
\[ \frac{-8x^2}{4x} = -2x \]
Add this to the quotient.

### Step 5: Multiply and subtract again
Multiply \( -2x \) by \( 4x + 1 \):
\[ -2x \cdot (4x + 1) = -8x^2 - 2x \]
Subtract this from \( -8x^2 + 10x + 3 \):
\[ (-8x^2 + 10x + 3) - (-8x^2 - 2x) = 10x + 2x + 3 = 12x + 3 \]

### Step 6: Final division
Now, divide \( 12x \) by \( 4x \):
\[ \frac{12x}{4x} = 3 \]
Add this to the quotient.

### Step 7: Multiply and find the remainder
Multiply \( 3 \) by \( 4x + 1 \):
\[ 3 \cdot (4x + 1) = 12x + 3 \]
Subtract this from \( 12x + 3 \):
\[ (12x + 3) - (12x + 3) = 0 \]

### Conclusion
The complete quotient is:
\[ 2x^2 - 2x + 3 \]
According to the problem, this is equal to \( 2x^2 + k \). Thus, we can compare:
\[ k = -2x + 3 \]
Since \( k \) is a constant, we take the constant term:
\[ k = 3 \]

### Final Answer
The value of \( k \) is \( 3 \).

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