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.