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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Would you like to go over any specific step in more detail?

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.