Use long division and synthetic division to find the quotient (you should not have a remainder). \( \begin{array}{ll}\text { 3. }\left(4 x^{3}-x^{2}-15 x+6\right) \div(x+2) & \text { (4) }\left(-3 x^{3}+7 x^{2}+27 x-28\right) \div(x-4) \\ \text { 5. }\left(9 x^{4}-3 x^{3}-20 x^{2}+24 x+32\right) \div(3 x+4) & \text { (6) }\left(-15 x^{5}+10 x^{4}+15 x^{2}-37 x+18\right) \div(3 x-2)\end{array} \)

Let's dive into the long division and synthetic division processes for these polynomial examples, starting with the first couple. For \( \left(4x^{3}-x^{2}-15x+6\right) \div (x+2) \): 1. **Long Division**: Divide the leading term \(4x^3\) by \(x\) to get \(4x^2\). 2. Multiply \(4x^2\) by \(x+2\) to get \(4x^3 + 8x^2\). Subtract this from the original polynomial: \( (-x^2 - 8x^2) + 6 \) becomes \( -9x^2 - 15x + 6 \). 3. Repeat the process: Divide \(-9x^2\) by \(x\) to get \(-9x\). Multiply and subtract again. 4. Continue until you've simplified completely: the quotient is \(4x^2 - 9x + 3\). For \( \left(-3x^{3}+7x^{2}+27x-28\right) \div (x-4) \): 1. Start by dividing \(-3x^3\) by \(x\) to get \(-3x^2\). 2. Multiply and subtract to find the next leading term: the process will yield \( -3x^2 + 19x - 3\). 3. Continue until all like terms are handled: the final quotient is \(-3x^2 - 5x + 9\). Now for synthetic division, we only focus on the coefficients and the zero of the divisor: For \( \left( 9x^4 - 3x^3 - 20x^2 + 24x + 32 \right) \div (3x + 4) \): 1. The zero of \(3x + 4\) is \( -\frac{4}{3} \). 2. Write down coefficients: \(9, -3, -20, 24, 32\). 3. Use synthetic division with \( -\frac{4}{3} \): Follow through with multiplication and adding down columns. 4. The result yields coefficients for the quotient: \(3x^3 - 5x^2 - 8x + 8\). Lastly, for \( \left(-15x^{5}+10x^{4}+15x^{2}-37x+18 \right) \div (3x - 2) \): 1. The zero is \( \frac{2}{3} \). 2. List out coefficients: \(-15, 10, 0, 15, -37, 18\). 3. Again, use synthetic division with \( \frac{2}{3} \) to find the final coefficients. 4. The result will show the quotient as \(-5x^4 + \frac{20}{3}x^3 + 10x^2 - 1x + 2\). And there you have it! Polynomial division made easy, like a math magician pulling numbers out of thin air! Remember to double-check your steps – even the best wizards can miscalculations!