To find the equation of the tangent line to the parabola $f(x) = x^2 + 2x + 2$ at the point $(2, 2)$, we need to follow these steps:

Step 1: Verify that the point lies on the curve

Substitute $x = 2$ into the function $f(x)$ to check if $f(2) = 2$.

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

But the given point is $(2, 2)$, and we calculated $f(2) = 10$, not 2. This means the point $(2, 2)$ is not on the parabola $f(x) = x^2 + 2x + 2$, so there is no tangent line at this point.

Alternative Approach

Since there was likely a miscommunication with the given point, I suggest verifying the point or function again. If you want to work through this problem with a different point on the curve, feel free to provide more details.

Do you have any questions or want to clarify the problem?

Here are 5 related questions:

1. How do you find the slope of a tangent line to a curve at a given point?
2. What happens if the point does not lie on the curve?
3. How can derivatives help in determining tangent lines?
4. Can a point outside the curve have a unique tangent line?
5. How can we generalize the formula for the equation of a tangent line?

Tip: When verifying whether a point is on a curve, always substitute the $x$-coordinate into the equation and check if the $y$-value matches.