To find $f(-4)$, we will substitute $x = -4$ into the function rule for $f(x)$:

$f(x) = 2x + 2x^2 + 10$

Now, substitute $x = -4$:

$f(-4) = 2(-4) + 2(-4)^2 + 10$

Step-by-step calculation:

1. $2(-4) = -8$
2. $(-4)^2 = 16$, so $2(16) = 32$
3. Finally, add $10$ to the result.

So,

$f(-4) = -8 + 32 + 10 = 34$

Thus, $f(-4) = 34$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you find $f(5)$ for the same function rule?
2. What is the general shape of the graph for this function?
3. How would $f(x)$ change if the $+10$ term were removed?
4. How do you solve for $f(x) = 0$ in this equation?
5. What would happen to $f(x)$ if you multiplied the entire equation by a constant?

Tip: Always calculate exponents before performing multiplication or addition in function rules!