Given the function $g(x) = x^2 + 4x + 1$, let's find the following values:

1. $g(2)$

Substitute $x = 2$ into the function:

$g(2) = (2)^2 + 4(2) + 1 = 4 + 8 + 1 = 13$

So, $g(2) = 13$.

2. $g(T)$

Substitute $x = T$ into the function:

$g(T) = T^2 + 4T + 1$

So, $g(T) = T^2 + 4T + 1$.

3. $g(x + 2)$

Substitute $x = x + 2$ into the function:

$g(x+2) = (x+2)^2 + 4(x+2) + 1$

First, expand $(x+2)^2$ and $4(x+2)$:

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

Now, combine all terms:

$g(x+2) = x^2 + 4x + 4 + 4x + 8 + 1 = x^2 + 8x + 13$

So, $g(x+2) = x^2 + 8x + 13$.

Summary:

1. $g(2) = 13$
2. $g(T) = T^2 + 4T + 1$
3. $g(x+2) = x^2 + 8x + 13$

Would you like further details or explanations on any of these steps?

Here are five related questions to expand your understanding:

1. What is $g(0)$ for the given function?
2. How would you find the roots of $g(x) = 0$?
3. What is the vertex of the parabola represented by $g(x)$?
4. How do you determine whether the parabola opens upwards or downwards?
5. What is the axis of symmetry for the function $g(x)$?

Tip: When substituting expressions into functions, ensure you fully expand and simplify the expression to avoid errors.