1. **State the problem:** We need to perform polynomial division for two expressions: a) $$\frac{8x^3 - 22x^2 + 11x + 6}{2x - 3}$$ b) $$\frac{4x^3 - 8x^2 + 11x - 4}{2x - 1}$$ 2. **Formula and rules:** Polynomial division is similar to long division with numbers. We divide the leading term of the numerator by the leading term of the denominator, multiply the entire divisor by this result, subtract, and repeat until the degree of the remainder is less than the divisor. 3. **Divide the first polynomial:** - Divide leading terms: $$\frac{8x^3}{2x} = 4x^2$$ - Multiply divisor by $$4x^2$$: $$4x^2(2x - 3) = 8x^3 - 12x^2$$ - Subtract: $$\left(8x^3 - 22x^2 + 11x + 6\right) - \left(8x^3 - 12x^2\right) = -10x^2 + 11x + 6$$ 4. **Next step:** - Divide leading terms: $$\frac{-10x^2}{2x} = -5x$$ - Multiply divisor by $$-5x$$: $$-5x(2x - 3) = -10x^2 + 15x$$ - Subtract: $$\left(-10x^2 + 11x + 6\right) - \left(-10x^2 + 15x\right) = -4x + 6$$ 5. **Next step:** - Divide leading terms: $$\frac{-4x}{2x} = -2$$ - Multiply divisor by $$-2$$: $$-2(2x - 3) = -4x + 6$$ - Subtract: $$\left(-4x + 6\right) - \left(-4x + 6\right) = 0$$ 6. **Result for first division:** $$4x^2 - 5x - 2$$ with remainder 0. 7. **Divide the second polynomial:** - Divide leading terms: $$\frac{4x^3}{2x} = 2x^2$$ - Multiply divisor by $$2x^2$$: $$2x^2(2x - 1) = 4x^3 - 2x^2$$ - Subtract: $$\left(4x^3 - 8x^2 + 11x - 4\right) - \left(4x^3 - 2x^2\right) = -6x^2 + 11x - 4$$ 8. **Next step:** - Divide leading terms: $$\frac{-6x^2}{2x} = -3x$$ - Multiply divisor by $$-3x$$: $$-3x(2x - 1) = -6x^2 + 3x$$ - Subtract: $$\left(-6x^2 + 11x - 4\right) - \left(-6x^2 + 3x\right) = 8x - 4$$ 9. **Next step:** - Divide leading terms: $$\frac{8x}{2x} = 4$$ - Multiply divisor by $$4$$: $$4(2x - 1) = 8x - 4$$ - Subtract: $$\left(8x - 4\right) - \left(8x - 4\right) = 0$$ 10. **Result for second division:** $$2x^2 - 3x + 4$$ with remainder 0. **Final answers:** - $$\frac{8x^3 - 22x^2 + 11x + 6}{2x - 3} = 4x^2 - 5x - 2$$ - $$\frac{4x^3 - 8x^2 + 11x - 4}{2x - 1} = 2x^2 - 3x + 4$$