1. **State the problem:** Divide the polynomial $$4x^2 - 6x + 8$$ by the binomial $$2x + 3$$ using polynomial long division. 2. **Recall the formula and rules:** Polynomial long division is similar to numerical long division. We divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by this quotient term, subtract from the dividend, and repeat with the remainder. 3. **Step 1:** Divide the leading term of the dividend $$4x^2$$ by the leading term of the divisor $$2x$$: $$\frac{4x^2}{2x} = 2x$$ 4. **Step 2:** Multiply the divisor $$2x + 3$$ by $$2x$$: $$2x(2x + 3) = 4x^2 + 6x$$ 5. **Step 3:** Subtract this from the dividend: $$\left(4x^2 - 6x + 8\right) - \left(4x^2 + 6x\right) = 4x^2 - 6x + 8 - 4x^2 - 6x = \cancel{4x^2} - 6x + 8 - \cancel{4x^2} - 6x = -12x + 8$$ 6. **Step 4:** Divide the new leading term $$-12x$$ by $$2x$$: $$\frac{-12x}{2x} = -6$$ 7. **Step 5:** Multiply the divisor by $$-6$$: $$-6(2x + 3) = -12x - 18$$ 8. **Step 6:** Subtract this from the previous remainder: $$(-12x + 8) - (-12x - 18) = -12x + 8 + 12x + 18 = \cancel{-12x} + 8 + \cancel{12x} + 18 = 26$$ 9. **Step 7:** Since the remainder $$26$$ is a constant and the divisor is degree 1, division stops here. 10. **Final answer:** $$\frac{4x^2 - 6x + 8}{2x + 3} = 2x - 6 + \frac{26}{2x + 3}$$