Let's analyze the given problem and solve it.

The function provided is $f(x) = 7x^2 + 3$, and you are asked to evaluate two expressions:

1. $f(x + 2)$
2. $f(x) + f(2)$

1. Evaluating $f(x + 2)$:

This means we need to substitute $x + 2$ into the function in place of $x$.

$f(x + 2) = 7(x + 2)^2 + 3$

First, expand $(x + 2)^2$:

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

Now, substitute this back into the function:

$f(x + 2) = 7(x^2 + 4x + 4) + 3$

Distribute the 7:

$f(x + 2) = 7x^2 + 28x + 28 + 3$

Combine like terms:

$f(x + 2) = 7x^2 + 28x + 31$

2. Evaluating $f(x) + f(2)$:

We already know $f(x) = 7x^2 + 3$, so now we need to find $f(2)$.

Substitute $2$ into the function:

$f(2) = 7(2)^2 + 3 = 7(4) + 3 = 28 + 3 = 31$

So, $f(x) + f(2)$ becomes:

$f(x) + f(2) = (7x^2 + 3) + 31 = 7x^2 + 34$

Final answers:

1. $f(x + 2) = 7x^2 + 28x + 31$
2. $f(x) + f(2) = 7x^2 + 34$

Would you like more details on any of these steps?

Here are five related questions to explore further:

1. How would the result change if the function were $f(x) = 7x^2 - 3$?
2. Can you explain why expanding $(x+2)^2$ results in $x^2 + 4x + 4$?
3. How do you find the vertex of the function $f(x) = 7x^2 + 28x + 31$?
4. What happens to $f(x) + f(a)$ for any value of $a$?
5. How would you solve $f(x + h)$ for a different linear shift like $h = 3$?

Tip: Always expand binomials like $(x + k)^2$ carefully by applying the formula $(x + k)^2 = x^2 + 2kx + k^2$.