Evaluate u square minus two for the function g of t equals to sqaure  root of t sqaure plus$$  $$ four t plus four.

To evaluate the expression $ u^2 - 2 $ for the function $ g(t) $, we first need to define $ u $ in terms of $ g(t) $. 

Given the function:
\[
g(t) = \sqrt{t^2 + 4t + 4}
\]

### Step 1: Simplify the function $ g(t) $

We notice that the expression under the square root is a perfect square:
\[
t^2 + 4t + 4 = (t + 2)^2
\]

So, the function simplifies to:
\[
g(t) = \sqrt{(t + 2)^2}
\]

Since $ \sqrt{(t + 2)^2} = |t + 2| $, we have:
\[
g(t) = |t + 2|
\]

### Step 2: Define $ u $

Let's assume that $ u = g(t) = |t + 2| $.

### Step 3: Evaluate $ u^2 - 2 $

Now, substitute $ u = |t + 2| $ into the expression $ u^2 - 2 $:
\[
u^2 - 2 = (|t + 2|)^2 - 2
\]

Since $ |t + 2|^2 = (t + 2)^2 $, the expression becomes:
\[
u^2 - 2 = (t + 2)^2 - 2
\]

Expanding $ (t + 2)^2 $, we get:
\[
(t + 2)^2 = t^2 + 4t + 4
\]

So, the expression is:
\[
u^2 - 2 = t^2 + 4t + 4 - 2
\]

Simplifying, we obtain:
\[
u^2 - 2 = t^2 + 4t + 2
\]

### Final Answer:

The expression $ u^2 - 2 $ for the function $ g(t) = \sqrt{t^2 + 4t + 4} $ evaluates to:
\[
t^2 + 4t + 2
\]

Would you like more details on any part of this solution, or do you have any questions?

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Here are 8 related questions:

1. What is the value of $ u^2 - 2 $ when $ t = 1 $?
2. How would the solution change if $ g(t) = \sqrt{t^2 + 4t + 5} $?
3. What is the domain of the function $ g(t) = \sqrt{t^2 + 4t + 4} $?
4. How do you determine if an expression under a square root is a perfect square?
5. What is the derivative of $ g(t) = \sqrt{t^2 + 4t + 4} $?
6. How can you graph the function $ g(t) = \sqrt{t^2 + 4t + 4} $?
7. What is the significance of the absolute value in $ g(t) = |t + 2| $?
8. How would you solve $ g(t) = 3 $ for $ t $?

**Tip:** When simplifying square roots, always check if the expression under the root is a perfect square. This can greatly simplify the function.

Learn how to evaluate the expression u^2 - 2 for the function g(t) = sqrt(t^2 + 4t + 4) step-by-step. Understand the concepts of square root functions and absolute values in mathematical expressions. Suitable for high school students.

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