Question 3 Use long division to divide $$ 2 x^{4}-3 x^{3}-24 x^{2}+76 x-60 $$ by $$ 2 x-3 $$. What is the new expression? $$ x^{3}+2 x^{2}-6 x+20 $$ $$ -x^{3}+6 x+20 $$ $$ 2 x^{3}-x^{2}-12 x+20 $$ $$ x^{3}-12 x+20 $$

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To divide $$ 2x^{4} - 3x^{3} - 24x^{2} + 76x - 60 $$ by $$ 2x - 3 $$ using long division, follow these steps:

1. **Set Up the Division:**
   $$
   \frac{2x^{4} - 3x^{3} - 24x^{2} + 76x - 60}{2x - 3}
   $$

2. **Divide the First Term:**
   - $$ \frac{2x^{4}}{2x} = x^{3} $$
   - Multiply $$ x^{3} $$ by $$ 2x - 3 $$:
     $$
     x^{3} \cdot (2x - 3) = 2x^{4} - 3x^{3}
     $$
   - Subtract from the original polynomial:
     $$
     (2x^{4} - 3x^{3} - 24x^{2} + 76x - 60) - (2x^{4} - 3x^{3}) = -24x^{2} + 76x - 60
     $$

3. **Next Division Step:**
   - $$ \frac{-24x^{2}}{2x} = -12x $$
   - Multiply $$ -12x $$ by $$ 2x - 3 $$:
     $$
     -12x \cdot (2x - 3) = -24x^{2} + 36x
     $$
   - Subtract:
     $$
     (-24x^{2} + 76x - 60) - (-24x^{2} + 36x) = 40x - 60
     $$

4. **Final Division Step:**
   - $$ \frac{40x}{2x} = 20 $$
   - Multiply $$ 20 $$ by $$ 2x - 3 $$:
     $$
     20 \cdot (2x - 3) = 40x - 60
     $$
   - Subtract:
     $$
     (40x - 60) - (40x - 60) = 0
     $$

5. **Conclusion:**
   The quotient is $$ x^{3} - 12x + 20 $$.

**Answer:**  
$$ x^{3}-12 x+20 $$
The new expression is $$ x^{3} - 12x + 20 $$.

Question 3 Use long division to divide \( 2 x^{4}-3 x^{3}-24 x^{2}+76 x-60 \) by \( 2 x-3 \). What is the new expression? \( x^{3}+2 x^{2}-6 x+20 \) \( -x^{3}+6 x+20 \) \( 2 x^{3}-x^{2}-12 x+20 \) \( x^{3}-12 x+20 \)

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