roblem 1. Let \( y=f(x)=-5 x^{2}+3 x+2 \). Find the Difference opoints) Quotient \( \frac{f(x+h)-f(x)}{h} \) and simplify. Detaild work:

To find the difference quotient \( \frac{f(x+h) - f(x)}{h} \) for the function \( f(x) = -5x^2 + 3x + 2 \), we will follow these steps: 1. **Calculate \( f(x+h) \)**: We need to substitute \( x+h \) into the function \( f(x) \). \[ f(x+h) = -5(x+h)^2 + 3(x+h) + 2 \] Expanding \( (x+h)^2 \): \[ (x+h)^2 = x^2 + 2xh + h^2 \] Therefore, \[ f(x+h) = -5(x^2 + 2xh + h^2) + 3(x+h) + 2 \] Now, distribute the terms: \[ f(x+h) = -5x^2 - 10xh - 5h^2 + 3x + 3h + 2 \] 2. **Calculate \( f(x+h) - f(x) \)**: Now we need to subtract \( f(x) \) from \( f(x+h) \). \[ f(x) = -5x^2 + 3x + 2 \] Thus, \[ f(x+h) - f(x) = (-5x^2 - 10xh - 5h^2 + 3x + 3h + 2) - (-5x^2 + 3x + 2) \] Simplifying this expression: \[ f(x+h) - f(x) = -10xh - 5h^2 + 3h \] 3. **Form the difference quotient**: Now we can form the difference quotient: \[ \frac{f(x+h) - f(x)}{h} = \frac{-10xh - 5h^2 + 3h}{h} \] We can simplify this by dividing each term by \( h \) (assuming \( h \neq 0 \)): \[ \frac{f(x+h) - f(x)}{h} = -10x - 5h + 3 \] 4. **Final Result**: The simplified form of the difference quotient is: \[ \frac{f(x+h) - f(x)}{h} = -10x - 5h + 3 \] This is the final answer for the difference quotient of the function \( f(x) = -5x^2 + 3x + 2 \).