The problem asks to find and simplify the expression $\frac{f(a+h) - f(a)}{h}$, given that $f(x) = 2x^2 - 1$.

Let's break down the steps:

1. Step 1: Find $f(a+h)$.

   • Substitute $a + h$ into $f(x)$: $f(a+h) = 2(a+h)^2 - 1$ Expand $(a+h)^2$: $f(a+h) = 2(a^2 + 2ah + h^2) - 1 = 2a^2 + 4ah + 2h^2 - 1$

2. Step 2: Find $f(a)$.

   • Substitute $a$ into $f(x)$: $f(a) = 2a^2 - 1$

3. Step 3: Substitute both expressions into $\frac{f(a+h) - f(a)}{h}$: [ \frac{f(a+h) - f(a)}{h} = \frac{(2a^