To solve the problem, we need to find the function $f(x)$ from the given equation and then use it to calculate $f(-7)$.

Step 1: Given Equation

The given equation is:

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

Step 2: Simplify the Equation

Combine like terms on the left-hand side:

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

Step 3: Solve for $f(x)$

Divide both sides by 3 to isolate $f(x)$:

$f(x) = x^2 + \frac{2}{3}$

Step 4: Find $f(-7)$

Now that we have the function $f(x)$, substitute $x = -7$ into the equation to find $f(-7)$:

$f(-7) = (-7)^2 + \frac{2}{3}$

$f(-7) = 49 + \frac{2}{3}$

Step 5: Simplify the Expression

To combine the terms:

$f(-7) = \frac{147}{3} + \frac{2}{3} = \frac{149}{3}$

So, $f(-7) = \frac{149}{3}$.

Final Answer:

The value of $f(-7)$ is $\boxed{\frac{149}{3}}$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would you determine the roots of the function $f(x) = x^2 + \frac{2}{3}$?
2. What is the derivative of the function $f(x) = x^2 + \frac{2}{3}$?
3. How does the graph of $f(x) = x^2 + \frac{2}{3}$ compare to the graph of $f(x) = x^2$?
4. What is the value of $f(2)$ using the function $f(x) = x^2 + \frac{2}{3}$?
5. How can you find the inverse of the function $f(x) = x^2 + \frac{2}{3}$?

Tip: When solving equations, always ensure that like terms are properly combined before isolating the variable.