g(x)=x2+4x+1 find following 1:g(2) 2:g(T) 3:(x+2) -x2+4x+1

Given the function \( g(x) = x^2 + 4x + 1 \), let's solve the following:

### 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. \( g(x + 2) - x^2 + 4x + 1 \)
First, we need to find \( g(x+2) \). As calculated earlier:

\[
g(x+2) = x^2 + 8x + 13
\]

Now, subtract \( x^2 + 4x + 1 \) from \( g(x+2) \):

\[
g(x+2) - (x^2 + 4x + 1) = (x^2 + 8x + 13) - (x^2 + 4x + 1)
\]

Simplify the expression by canceling like terms:

\[
= x^2 + 8x + 13 - x^2 - 4x - 1 = 4x + 12
\]

So, \( g(x+2) - (x^2 + 4x + 1) = 4x + 12 \).

### Summary:
1. \( g(2) = 13 \)
2. \( g(T) = T^2 + 4T + 1 \)
3. \( g(x+2) - x^2 + 4x + 1 = 4x + 12 \)

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

Here are five related questions to consider:

1. What is the derivative of \( g(x) = x^2 + 4x + 1 \) with respect to \( x \)?
2. How would you solve \( g(x) = 0 \) for \( x \)?
3. What does the expression \( g(x+2) \) represent geometrically?
4. How does the subtraction \( g(x+2) - x^2 + 4x + 1 \) relate to the original function?
5. Can \( g(x) = x^2 + 4x + 1 \) be factored, and if so, how?

**Tip:** When working with functions, always carefully expand and simplify expressions step by step to avoid mistakes.

Learn how to evaluate g(x) = x^2 + 4x + 1 for specific values, including g(2) and g(T), and simplify expressions involving function composition and subtraction. This problem covers key concepts in quadratic equations and function operations. Suitable for high school students.

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