To expand the expression \((6x + 4)(-2x^2 - 3x + 1)\), we use the distributive property (also known as FOIL for binomials), multiplying each term in the first expression by each term in the second expression.

Let's break it down:

\[
(6x + 4)(-2x^2 - 3x + 1)
\]

### Step 1: Multiply \(6x\) by each term in \(-2x^2 - 3x + 1\)
\[
6x \times (-2x^2) = -12x^3
\]
\[
6x \times (-3x) = -18x^2
\]
\[
6x \times (1) = 6x
\]

### Step 2: Multiply \(4\) by each term in \(-2x^2 - 3x + 1\)
\[
4 \times (-2x^2) = -8x^2
\]
\[
4 \times (-3x) = -12x
\]
\[
4 \times (1) = 4
\]

### Step 3: Combine all the terms
Now add all the results from steps 1 and 2:
\[
-12x^3 - 18x^2 + 6x - 8x^2 - 12x + 4
\]

### Step 4: Combine like terms
\[
-12x^3 + (-18x^2 - 8x^2) + (6x - 12x) + 4
\]
\[
= -12x^3 - 26x^2 - 6x + 4
\]

Thus, the expanded form is:
\[
\boxed{-12x^3 - 26x^2 - 6x + 4}
\]

Would you like further explanation or have any questions?

Here are some related questions for practice:

1. Expand \((3x + 2)(x^2 - 4x + 5)\).
2. Simplify the expression \((5x - 3)(-x^2 + 2x - 1)\).
3. Factor the expression \(-12x^3 - 26x^2 - 6x + 4\).
4. Solve for \(x\) in \(-12x^3 - 26x^2 - 6x + 4 = 0\).
5. Expand and simplify \((7x + 1)(-x^2 + x + 6)\).

**Tip**: When multiplying polynomials, make sure to carefully distribute each term to avoid missing any parts of the product.

Learn how to expand and simplify the expression (6x + 4)(-2x^2 - 3x + 1) using the distributive property. This problem covers polynomial multiplication and algebraic techniques, making it suitable for students in Grades 9-11.

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