We are given the function:

$f(x) = 4x^2 + 4x - 2$

We need to evaluate $f(x+h)$, which means substituting $x+h$ for $x$ in the function.

Step 1: Substitute $x+h$ into $f(x)$

$f(x+h) = 4(x+h)^2 + 4(x+h) - 2$

Step 2: Expand the Squared Term

$(x+h)^2 = x^2 + 2xh + h^2$

$4(x+h)^2 = 4(x^2 + 2xh + h^2) = 4x^2 + 8xh + 4h^2$

Step 3: Expand the Linear Term

$4(x+h) = 4x + 4h$

Step 4: Combine All Terms

$f(x+h) = 4x^2 + 8xh + 4h^2 + 4x + 4h - 2$

This is the simplified expression for $f(x+h)$.

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Would you like me to expand further or clarify any part of the process?

Here are some related questions to deepen understanding:

1. How do we compute the difference quotient $\frac{f(x+h) - f(x)}{h}$ for this function?
2. What is the derivative $f'(x)$ using the definition of a derivative?
3. How does substituting $h = 0$ in $f(x+h)$ verify $f(x)$?
4. How can we use $f(x+h)$ to approximate values of $f(x)$ for small $h$?
5. What happens to $f(x+h)$ as $h \to 0$, and why is this important in calculus?

Tip:

When expanding $(x+h)^2$, always remember the middle term $2xh$, as it's commonly missed in algebraic expansions!