Polynomial long division: Problem type 3 Divide. (-4 x^{4}+3 x^{2}+12-11 x^{3}) /(-x^{2}-2 x+3) Write your answer in the following form: Quotient +frac{Remainder }{-x^{2}-2 x+3}

To divide \(-4x^{4} + 3x^{2} + 12 - 11x^{3}\) by \(-x^{2} - 2x + 3\), we start by reordering the polynomial in standard form: \[ -4x^{4} - 11x^{3} + 3x^{2} + 12 \] Now, we perform polynomial long division. First, we divide the leading term of the dividend \(-4x^{4}\) by the leading term of the divisor \(-x^{2}\) to get \(4x^2\). Next, we multiply the entire divisor \(-x^2 - 2x + 3\) by \(4x^2\) to get: \[ 4x^2(-x^2 - 2x + 3) = 4x^4 + 8x^3 - 12x^2 \] Subtract this from the original polynomial: \[ (-4x^4 - 11x^3 + 3x^2 + 12) - (4x^4 + 8x^3 - 12x^2) = -19x^3 + 15x^2 + 12 \] Now, we repeat the process: divide \(-19x^3\) by \(-x^2\) to get \(19x\). Next, multiply the divisor by \(19x\): \[ 19x(-x^2 - 2x + 3) = -19x^3 - 38x^2 + 57x \] Subtract again: \[ (-19x^3 + 15x^2 + 12) - (-19x^3 - 38x^2 + 57x) = 53x^2 - 57x + 12 \] Now we divide \(53x^2\) by \(-x^2\) to get \(-53\). Multiply the divisor by \(-53\): \[ -53(-x^2 - 2x + 3) = 53x^2 + 106x - 159 \] Subtract: \[ (53x^2 - 57x + 12) - (53x^2 + 106x - 159) = -163x + 171 \] At this point, our degree of the remainder \(-163x + 171\) is less than the degree of the divisor, so we stop here. Thus, the final result in the desired form is: \[ 4x^2 + 19x - 53 + \frac{-163x + 171}{-x^2 - 2x + 3} \]