Let's simplify and factorize the given expression step by step:

The expression is:

\[
(9x^2 + 12x + 4) - 2x(3x + 2) + (4 - 9x^2)
\]

### Step 1: Expand the terms
First, let's distribute the terms in the second part of the expression.

\[
2x(3x + 2) = 6x^2 + 4x
\]

Now, the expression becomes:

\[
(9x^2 + 12x + 4) - (6x^2 + 4x) + (4 - 9x^2)
\]

### Step 2: Combine like terms
Let's combine the like terms from each part of the expression.

- \(9x^2 - 6x^2 - 9x^2 = -6x^2\)
- \(12x - 4x = 8x\)
- \(4 + 4 = 8\)

So the expression simplifies to:

\[
-6x^2 + 8x + 8
\]

### Step 3: Factorize the simplified expression
Now, factor out the greatest common factor (GCF), which is \(-2\).

\[
-2(3x^2 - 4x - 4)
\]

Next, we factorize the quadratic \(3x^2 - 4x - 4\).

We can apply the quadratic factoring method. Find two numbers that multiply to \(3 \times -4 = -12\) and add to \(-4\). These numbers are \(-6\) and \(2\).

So, rewrite the middle term as:

\[
3x^2 - 6x + 2x - 4
\]

Group the terms:

\[
(3x^2 - 6x) + (2x - 4)
\]

Factor each group:

\[
3x(x - 2) + 2(x - 2)
\]

Now factor out the common binomial factor \((x - 2)\):

\[
(3x + 2)(x - 2)
\]

### Step 4: Final factorized form
Substitute this back into the original expression:

\[
-2(3x + 2)(x - 2)
\]

So, the fully factorized form of the expression is:

\[
\boxed{-2(3x + 2)(x - 2)}
\]

Would you like further clarification or details? Here are five related questions to explore:

1. How do you factor quadratics using the grouping method?
2. What are the general steps to simplify complex algebraic expressions?
3. How can you check if your factorization is correct?
4. Can all quadratic expressions be factored over real numbers?
5. How would you solve the equation \(-2(3x + 2)(x - 2) = 0\)?

**Tip:** Always combine like terms first before attempting to factor an expression.

factorise (9x^2 + 12x + 4) - 2x(3x + 2) + (4 - 9x^2)

Learn how to factorize the expression (9x^2 + 12x + 4) - 2x(3x + 2) + (4 - 9x^2) step by step. This solution demonstrates key algebraic concepts such as combining like terms and quadratic factorization. Suitable for students in Grades 9-12.

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