College Algebra: Test \#2 1. Find and simplify the difference quotient for for \( f(x)=\frac{-5 x^{2}}{2}-3 x \). Show all work.

To find the difference quotient for the function \( f(x) = \frac{-5x^{2}}{2} - 3x \), we start by calculating \( f(x+h) \): 1. **Calculate \( f(x+h) \)**: \[ f(x+h) = \frac{-5(x+h)^{2}}{2} - 3(x+h) \] Expanding \( (x+h)^{2} \): \[ = \frac{-5(x^{2} + 2xh + h^{2})}{2} - 3x - 3h \] Simplifying: \[ = \frac{-5x^{2} - 10xh - 5h^{2}}{2} - 3x - 3h \] 2. **Set up the difference quotient**: \[ \frac{f(x+h) - f(x)}{h} \] Substitute \( f(x+h) \) and \( f(x) \): \[ = \frac{\left(\frac{-5x^{2} - 10xh - 5h^{2}}{2} - 3x - 3h\right) - \left(\frac{-5x^{2}}{2} - 3x\right)}{h} \] Simplifying further, the \( -3x \) terms cancel out: \[ = \frac{\left(\frac{-5x^{2} - 10xh - 5h^{2}}{2} + \frac{5x^{2}}{2} - 3h\right)}{h} \] 3. **Combine like terms**: \[ = \frac{\left(-10xh - 5h^{2} - 6h\right)}{2h} \] Factor \( h \) out of the numerator: \[ = \frac{h(-10x - 5h - 6)}{2h} = \frac{-10x - 5h - 6}{2} \] 4. **Final result**: The simplified difference quotient is: \[ \frac{-10x - 5h - 6}{2} \] And there you have it—a clear-cut answer for your difference quotient, complete with all the steps!