Divide. \[ \left(8 x^{3}-19 x^{2}+20 x^{4}-6+2 x\right) \div\left(-4 x^{2}+3\right) \] Write your answer in the following form: Quotient \( +\frac{\text { Remainder }}{-4 x^{2}+3} \). \( \frac{8 x^{3}-19 x^{2}+20 x^{4}-6+2 x}{-4 x^{2}+3}=\square+\frac{\square}{-4 x^{2}+3} \)

Let's start by rewriting the polynomial in standard form to make the division easier: \[ 20x^4 + 8x^3 - 19x^2 + 2x - 6 \] Now we will perform polynomial long division between \( 20x^4 + 8x^3 - 19x^2 + 2x - 6 \) and \( -4x^2 + 3 \). 1. **First Division**: Divide the leading term of the dividend \( 20x^4 \) by the leading term of the divisor \( -4x^2 \): \[ \frac{20x^4}{-4x^2} = -5x^2 \] Multiply \( -5x^2 \) by the entire divisor: \[ -5x^2(-4x^2 + 3) = 20x^4 - 15x^2 \] Subtract this from the original polynomial: \[ (20x^4 + 8x^3 - 19x^2 + 2x - 6) - (20x^4 - 15x^2) = 8x^3 - 4x^2 + 2x - 6 \] 2. **Second Division**: Divide the leading term \( 8x^3 \) by the leading term \( -4x^2 \): \[ \frac{8x^3}{-4x^2} = -2x \] Multiply \( -2x \) by the entire divisor: \[ -2x(-4x^2 + 3) = 8x^3 - 6x \] Subtract this from the current polynomial: \[ (8x^3 - 4x^2 + 2x - 6) - (8x^3 - 6x) = -4x^2 + 8x - 6 \] 3. **Third Division**: Divide \( -4x^2 \) by \( -4x^2 \): \[ \frac{-4x^2}{-4x^2} = 1 \] Multiply \( 1 \) by the entire divisor: \[ 1(-4x^2 + 3) = -4x^2 + 3 \] Subtract from the current polynomial: \[ (-4x^2 + 8x - 6) - (-4x^2 + 3) = 8x - 9 \] Now we've completed the division, and the result can be written as follows: \[ \frac{8 x^{3}-19 x^{2}+20 x^{4}-6+2 x}{-4 x^{2}+3} = -5x^2 - 2x + 1 + \frac{8x - 9}{-4x^2 + 3} \] Thus, we can finalize our answer: \[ \frac{8 x^{3}-19 x^{2}+20 x^{4}-6+2 x}{-4 x^{2}+3} = -5x^2 - 2x + 1 + \frac{8x - 9}{-4 x^{2}+3} \]