To factorize the given expression \( 5 - (3a^2 - 2a)(6 - 3a^2 + 2a) \), we'll follow these steps:

### Step 1: Expand the product
We first expand the product \( (3a^2 - 2a)(6 - 3a^2 + 2a) \) using the distributive property (also known as FOIL for binomials).

\[
(3a^2 - 2a)(6 - 3a^2 + 2a) = 3a^2(6) + 3a^2(-3a^2) + 3a^2(2a) - 2a(6) - 2a(-3a^2) - 2a(2a)
\]

Now simplify each term:

\[
= 18a^2 - 9a^4 + 6a^3 - 12a + 6a^3 - 4a^2
\]

Combine like terms:

\[
= -9a^4 + 12a^3 + 14a^2 - 12a
\]

### Step 2: Substitute the expanded result into the original expression
Now, substitute the expanded form into the original expression:

\[
5 - (-9a^4 + 12a^3 + 14a^2 - 12a)
\]

Distribute the minus sign:

\[
= 5 + 9a^4 - 12a^3 - 14a^2 + 12a
\]

### Step 3: Final expression
The expanded and simplified expression is:

\[
9a^4 - 12a^3 - 14a^2 + 12a + 5
\]

This is the factored form of the original expression, as it cannot be factored further into simple terms.

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Here are 5 related questions for further practice:
1. How do you factor a quadratic expression?
2. What are common methods to factorize polynomials of higher degrees?
3. Can you factor the expression \( 6a^2 - 5a - 4 \)?
4. What is the process of expanding and simplifying binomials?
5. How do you use the distributive property to simplify expressions?

**Tip:** When factorizing, always look for common factors or patterns such as difference of squares or perfect square trinomials.

Learn how to factorize the expression 5 - (3a^2 - 2a)(6 - 3a^2 + 2a) by expanding, simplifying, and combining like terms. This step-by-step guide explains key algebraic concepts, including polynomial expansion and simplification, suitable for Grades 10-12.

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