To expand the expression \((−2x−1)(x^2+9x+3)\), we will use the distributive property, which involves multiplying each term in the first expression by each term in the second expression.

Here's the step-by-step process:

\[
(-2x - 1)(x^2 + 9x + 3)
\]

### Step 1: Distribute \(-2x\) across each term in \((x^2 + 9x + 3)\):
\[
-2x \cdot x^2 = -2x^3
\]
\[
-2x \cdot 9x = -18x^2
\]
\[
-2x \cdot 3 = -6x
\]

### Step 2: Distribute \(-1\) across each term in \((x^2 + 9x + 3)\):
\[
-1 \cdot x^2 = -x^2
\]
\[
-1 \cdot 9x = -9x
\]
\[
-1 \cdot 3 = -3
\]

### Step 3: Combine all terms:
\[
-2x^3 - 18x^2 - 6x - x^2 - 9x - 3
\]

### Step 4: Simplify by combining like terms:
\[
-2x^3 - 19x^2 - 15x - 3
\]

So, the expanded form of \((−2x−1)(x^2+9x+3)\) is:

\[
\boxed{-2x^3 - 19x^2 - 15x - 3}
\]

Would you like more details or have any questions about this process?

Here are 5 related questions for further exploration:
1. How do you use the distributive property with more than two terms in each factor?
2. What is the standard form of a polynomial?
3. How can you factor \(-2x^3 - 19x^2 - 15x - 3\) back into its original form?
4. What are the roots of the polynomial \(-2x^3 - 19x^2 - 15x - 3\)?
5. How do you determine the degree of a polynomial?

**Tip:** When expanding expressions using the distributive property, carefully align like terms to make combining them easier.

Learn how to expand and simplify the polynomial expression (−2x−1)(x^2 + 9x + 3) using the distributive property. This step-by-step solution covers key algebra concepts, making it ideal for students in Grades 9-10.

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