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The two parts of the question involve polynomial multiplication. Let's solve them:

### Part (a): 
\[
(3x - 1)(x^2 - x - 1)
\]

We'll use the distributive property (also called FOIL method for binomials):

\[
= 3x(x^2 - x - 1) - 1(x^2 - x - 1)
\]

Distribute the terms:

\[
= 3x \cdot x^2 - 3x \cdot x - 3x \cdot 1 - (x^2 - x - 1)
\]
\[
= 3x^3 - 3x^2 - 3x - x^2 + x + 1
\]

Now combine like terms:

\[
= 3x^3 - 4x^2 - 2x + 1
\]

So, the simplified result for part (a) is:

\[
3x^3 - 4x^2 - 2x + 1
\]

### Part (b):
\[
(a^2 + 2a + 2)(3a^2 + 4a + 4)
\]

Again, we'll use the distributive property to multiply:

\[
= a^2(3a^2 + 4a + 4) + 2a(3a^2 + 4a + 4) + 2(3a^2 + 4a + 4)
\]

Distribute each term:

\[
= a^2 \cdot 3a^2 + a^2 \cdot 4a + a^2 \cdot 4 + 2a \cdot 3a^2 + 2a \cdot 4a + 2a \cdot 4 + 2 \cdot 3a^2 + 2 \cdot 4a + 2 \cdot 4
\]
\[
= 3a^4 + 4a^3 + 4a^2 + 6a^3 + 8a^2 + 8a + 6a^2 + 8a + 8
\]

Now combine like terms:

\[
= 3a^4 + (4a^3 + 6a^3) + (4a^2 + 8a^2 + 6a^2) + (8a + 8a) + 8
\]
\[
= 3a^4 + 10a^3 + 18a^2 + 16a + 8
\]

So, the simplified result for part (b) is:

\[
3a^4 + 10a^3 + 18a^2 + 16a + 8
\]

Would you like more details on these solutions or have any other questions?

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Here are some relative questions:
1. How can you apply the distributive property in more complex polynomial multiplication?
2. What strategies can simplify combining like terms?
3. Can you expand and factor the results further?
4. How do different methods, like the box method or grid method, compare to the distributive approach?
5. Why is it important to arrange terms in standard form?

**Tip:** When multiplying polynomials, always align terms with the same powers to simplify combining them!

Perform the following multiplications and simplify your results: (a) (3x - 1)(x^2 - x - 1), (b) (a^2 + 2a + 2)(3a^2 + 4a + 4)

Learn how to simplify polynomial multiplications, including (3x - 1)(x^2 - x - 1) and (a^2 + 2a + 2)(3a^2 + 4a + 4). This step-by-step solution uses the distributive property and combines like terms for Grades 9-11 students.

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