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The problem asks you to find and simplify expressions involving the product of two functions \( f(x) \) and \( g(x) \), where:

\[
f(x) = x + 2 \quad \text{and} \quad g(x) = x^2 + x + 1
\]

Let's solve each part step by step.

### Part (a): \( (fg)(x) \)
The product of two functions, \( (fg)(x) \), is defined as the product of \( f(x) \) and \( g(x) \):

\[
(fg)(x) = f(x) \cdot g(x)
\]

Substitute the given expressions for \( f(x) \) and \( g(x) \):

\[
(fg)(x) = (x + 2) \cdot (x^2 + x + 1)
\]

Now expand the product:

\[
(x + 2)(x^2 + x + 1) = x(x^2 + x + 1) + 2(x^2 + x + 1)
\]

\[
= x^3 + x^2 + x + 2x^2 + 2x + 2
\]

Now combine like terms:

\[
= x^3 + 3x^2 + 3x + 2
\]

Thus, \( (fg)(x) = x^3 + 3x^2 + 3x + 2 \).

### Part (b): \( (fg)(-2) \)
For \( (fg)(-2) \), substitute \( x = -2 \) into the simplified expression from part (a):

\[
(fg)(-2) = (-2)^3 + 3(-2)^2 + 3(-2) + 2
\]

Now compute the powers and products:

\[
= -8 + 3(4) + 3(-2) + 2
\]
\[
= -8 + 12 - 6 + 2
\]
\[
= 0
\]

Thus, \( (fg)(-2) = 0 \).

### Part (c): \( (fg)(1) \)
For \( (fg)(1) \), substitute \( x = 1 \) into the simplified expression from part (a):

\[
(fg)(1) = (1)^3 + 3(1)^2 + 3(1) + 2
\]

Now compute:

\[
= 1 + 3(1) + 3 + 2
\]
\[
= 1 + 3 + 3 + 2
\]
\[
= 9
\]

Thus, \( (fg)(1) = 9 \).

### Summary of Results:
- \( (fg)(x) = x^3 + 3x^2 + 3x + 2 \)
- \( (fg)(-2) = 0 \)
- \( (fg)(1) = 9 \)

Would you like further details on any part? Here are some related questions:

1. What is the general method for multiplying two polynomials?
2. How can you factor polynomials like the one in \( (fg)(x) \)?
3. How does function composition differ from function multiplication?
4. Can you apply the product of functions to solve real-world problems?
5. What are the domain and range of \( f(x) \cdot g(x) \)?

Tip: Always expand products step-by-step to avoid missing terms when simplifying polynomial expressions.

Let f(x) = x + 2 and g(x) = x^2 + x + 1. Find (fg)(x), (fg)(-2), and (fg)(1).

Learn how to multiply two polynomial functions, f(x) = x + 2 and g(x) = x^2 + x + 1, step by step. This problem involves algebraic expansion and simplification of polynomials. Includes solutions for (fg)(x), (fg)(-2), and (fg)(1). Suitable for Grades 10-12.

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