Let's start by analyzing and evaluating each part of the given expressions.

Function: $f(x) = -6x^2 - 2x + 5$

1. Evaluate $f(x+h)$:

To find $f(x+h)$, we need to substitute $x+h$ into the function in place of $x$.

$f(x+h) = -6(x+h)^2 - 2(x+h) + 5$

Now expand and simplify:

• Expand $(x+h)^2 = x^2 + 2xh + h^2$: $f(x+h) = -6(x^2 + 2xh + h^2) - 2(x+h) + 5$
• Distribute: $f(x+h) = -6x^2 - 12xh - 6h^2 - 2x - 2h + 5$

This is the simplified form of $f(x+h)$:

$f(x+h) = -6x^2 - 12xh - 6h^2 - 2x - 2h + 5$

2. Evaluate $\frac{f(x+h) - f(x)}{h}$:

Now, let's calculate the difference $f(x+h) - f(x)$ and then divide by $h$.

[ \frac{f(x+h) - f(x)}{h} = \frac{[-6x^2 - 12xh - 6h^2 - 2x - 2h + 5] - [-6x^2 - 2x + 5]}{h} ]

Simplify the numerator:

• Combine like terms: [ f(x+h) - f(x) = [-6x^2 - 12xh - 6h^2 - 2x - 2h + 5] - [-6x^2 - 2x + 5] ] $= -12xh - 6h^2 - 2h$

Now, divide the entire expression by $h$:

$\frac{f(x+h) - f(x)}{h} = \frac{-12xh - 6h^2 - 2h}{h}$

Factor out $h$ from the numerator:

$\frac{-12xh - 6h^2 - 2h}{h} = -12x - 6h - 2$

So, the final simplified form of $\frac{f(x+h) - f(x)}{h}$ is:

$\frac{f(x+h) - f(x)}{h} = -12x - 6h - 2$

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Would you like any further details or have questions?

Here are 5 related questions to expand your understanding:

1. What is the difference between $f(x+h)$ and $f(x)$ geometrically?
2. How does the expression $\frac{f(x+h) - f(x)}{h}$ relate to the concept of a derivative?
3. Can you derive the general formula for the derivative of $f(x) = ax^2 + bx + c$?
4. How would you apply this method to evaluate $f'(x)$ for more complex functions?
5. What does the term $h$ approaching zero signify in calculus?

Tip: The expression $\frac{f(x+h) - f(x)}{h}$ is the difference quotient, which is fundamental in finding the derivative of a function.