What is the remainder when doing the polynomial division shown below? (-6x^3+5x^2+18x-11)/(3x+2) .

To find the remainder of the polynomial division, we can use long division or synthetic division. Here, we'll use long division.<br /><br />1. Divide the leading term of the dividend by the leading term of the divisor: \(\frac{-6x^3}{3x} = -2x^2\).<br />2. Multiply the entire divisor by this result: \((-2x^2)(3x + 2) = -6x^3 - 4x^2\).<br />3. Subtract this from the original dividend: \((-6x^3 + 5x^2 + 18x - 11) - (-6x^3 - 4x^2) = 9x^2 + 18x - 11\).<br />4. Repeat the process with the new polynomial:<br /> - Divide the leading term: \(\frac{9x^2}{3x} = 3x\).<br /> - Multiply the entire divisor: \((3x)(3x + 2) = 9x^2 + 6x\).<br /> - Subtract: \((9x^2 + 18x - 11) - (9x^2 + 6x) = 12x - 11\).<br />5. Repeat the process again:<br /> - Divide the leading term: \(\frac{12x}{3x} = 4\).<br /> - Multiply the entire divisor: \((4)(3x + 2) = 12x + 8\).<br /> - Subtract: \((12x - 11) - (12x + 8) = -19\).<br /><br />The remainder is \(-19\).