Write the quotient and remainder when we divide :
`(8x^(4) + 10x^(3) - 5x^(2) - 4 x + 1)` by `(2x^(2) + x - 1)`

Answer

To find the quotient and remainder when dividing the polynomial \(8x^4 + 10x^3 - 5x^2 - 4x + 1\) by \(2x^2 + x - 1\), we can use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set Up the Division
We want to divide \(8x^4 + 10x^3 - 5x^2 - 4x + 1\) by \(2x^2 + x - 1\).

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

### Step 3: Multiply and Subtract
Now, multiply \(4x^2\) by the entire divisor \(2x^2 + x - 1\):
\[ 4x^2(2x^2 + x - 1) = 8x^4 + 4x^3 - 4x^2 \]
Subtract this from the original polynomial:
\[ (8x^4 + 10x^3 - 5x^2 - 4x + 1) - (8x^4 + 4x^3 - 4x^2) = (10x^3 - 4x^3) + (-5x^2 + 4x^2) - 4x + 1 \]
This simplifies to:
\[ 6x^3 - x^2 - 4x + 1 \]

### Step 4: Repeat the Process
Now, repeat the process with the new polynomial \(6x^3 - x^2 - 4x + 1\).

Divide the leading term \(6x^3\) by \(2x^2\):
\[ \frac{6x^3}{2x^2} = 3x \]
This is the next term of the quotient.

Multiply \(3x\) by the divisor:
\[ 3x(2x^2 + x - 1) = 6x^3 + 3x^2 - 3x \]
Subtract this from \(6x^3 - x^2 - 4x + 1\):
\[ (6x^3 - x^2 - 4x + 1) - (6x^3 + 3x^2 - 3x) = (-x^2 - 3x^2) + (-4x + 3x) + 1 \]
This simplifies to:
\[ -4x^2 - x + 1 \]

### Step 5: Continue the Process
Now, divide \(-4x^2\) by \(2x^2\):
\[ \frac{-4x^2}{2x^2} = -2 \]
This is the next term of the quotient.

Multiply \(-2\) by the divisor:
\[ -2(2x^2 + x - 1) = -4x^2 - 2x + 2 \]
Subtract this from \(-4x^2 - x + 1\):
\[ (-4x^2 - x + 1) - (-4x^2 - 2x + 2) = (-x + 2x) + (1 - 2) \]
This simplifies to:
\[ x - 1 \]

### Step 6: Final Result
Now, we cannot divide further since the degree of the remainder \(x - 1\) is less than the degree of the divisor \(2x^2 + x - 1\).

### Conclusion
The quotient is:
\[ 4x^2 + 3x - 2 \]
And the remainder is:
\[ x - 1 \]