Find and simplify \[ \frac{(a+h)-f(a)}{h},(h \neq 0) \] for the following function. \[ f(x)=4 x^{2}-2 x+7 \] -2

To simplify the expression \[ \frac{(a+h)-f(a)}{h} \] for the function \[ f(x) = 4x^2 - 2x + 7, \] we will follow these steps: 1. **Calculate \( f(a) \)**: Substitute \( a \) into the function \( f(x) \). 2. **Substitute \( f(a) \) into the expression**: Replace \( f(a) \) in the expression. 3. **Simplify the expression**: Perform algebraic simplifications. ### Step 1: Calculate \( f(a) \) Substituting \( a \) into the function: \[ f(a) = 4a^2 - 2a + 7. \] ### Step 2: Substitute \( f(a) \) into the expression Now, we substitute \( f(a) \) into the expression: \[ \frac{(a+h) - f(a)}{h} = \frac{(a+h) - (4a^2 - 2a + 7)}{h}. \] ### Step 3: Simplify the expression Now, we simplify the numerator: \[ a + h - (4a^2 - 2a + 7) = a + h - 4a^2 + 2a - 7 = -4a^2 + 3a + h - 7. \] Thus, the expression becomes: \[ \frac{-4a^2 + 3a + h - 7}{h}. \] Now, we can separate the terms in the numerator: \[ \frac{-4a^2 + 3a - 7}{h} + \frac{h}{h} = \frac{-4a^2 + 3a - 7}{h} + 1. \] So, the final simplified expression is: \[ \frac{-4a^2 + 3a - 7}{h} + 1. \] This is the simplified form of the given expression.