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? \( \begin{array}{l}\text { 2 } x^{3}+2 x^{2}-6 x+20 \\ -x^{3}-12 x+20 \\ x^{3}-12 x+20\end{array} \)

To carry out long division for the polynomial \( 2x^{4} - 3x^{3} - 24x^{2} + 76x - 60 \) divided by \( 2x - 3 \), let's proceed step-by-step: 1. **First Division**: Divide the leading term \( 2x^4 \) by the leading term \( 2x \) to get \( x^3 \). Multiply \( x^3 \) by \( 2x - 3 \) to get \( 2x^4 - 3x^3 \). 2. **Subtract**: Subtract \( (2x^4 - 3x^3) \) from the original polynomial: \[ (2x^{4} - 3x^{3}) - (2x^{4} - 3x^{3}) = 0 \] The new polynomial is \( -24x^{2} + 76x - 60 \). 3. **Second Division**: Next, divide \( -24x^{2} \) by \( 2x \), giving \( -12x \). Multiply \( -12x \) by \( 2x - 3 \) to get \( -24x^{2} + 36x \). 4. **Subtract Again**: Now subtract: \[ (-24x^{2} + 76x) - (-24x^{2} + 36x) = 40x \] The new polynomial is \( 40x - 60 \). 5. **Third Division**: Divide \( 40x \) by \( 2x \) to get \( 20 \). Multiply \( 20 \) by \( 2x - 3 \) to get \( 40x - 60 \). 6. **Final Subtraction**: Subtract: \[ (40x - 60) - (40x - 60) = 0 \] Since there’s nothing left at the end, the division is exact! The final expression you obtain from the division is: \[ \boxed{x^{3} - 12x + 20} \]