For the function 
f(x)=-6x2-2x+5
, evaluate and fully simplify each of the following.

f(x+h)
 = 
  


f(x+h)-f(x)h
 = 



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:

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

---

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.

For the function f(x) = -6x^2 - 2x + 5, evaluate and fully simplify each of the following: f(x+h), and (f(x+h) - f(x)) / h.

Explore how to evaluate and simplify f(x+h) and the difference quotient for the quadratic function f(x) = -6x^2 - 2x + 5. This guide walks through expanding and simplifying polynomial expressions and calculating the difference quotient, a key concept in calculus. Suitable for high school students in Grades 9-12.

Math Problem Statement

f(x+h)

f(x+h)-f(x)h

Solution

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