Expand the expression to a polynomial in standard form:
left parenthesis, x, squared, minus, 2, x, minus, 1, right parenthesis, left parenthesis, 2, x, squared, plus, 4, x, minus, 6, right parenthesis
(x 
2
 −2x−1)(2x 
2
 +4x−6)

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

Let's break it down step by step:

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

1. Multiply \(x^2\) by each term in the second polynomial:

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

2. Multiply \(-2x\) by each term in the second polynomial:

\[
-2x \cdot 2x^2 = -4x^3
\]
\[
-2x \cdot 4x = -8x^2
\]
\[
-2x \cdot (-6) = 12x
\]

3. Multiply \(-1\) by each term in the second polynomial:

\[
-1 \cdot 2x^2 = -2x^2
\]
\[
-1 \cdot 4x = -4x
\]
\[
-1 \cdot (-6) = 6
\]

Now, let's combine all these terms:

\[
2x^4 + 4x^3 - 6x^2 - 4x^3 - 8x^2 + 12x - 2x^2 - 4x + 6
\]

Next, we combine like terms:

- For \(x^4\): \(2x^4\) (no like terms)
- For \(x^3\): \(4x^3 - 4x^3 = 0\) (these terms cancel out)
- For \(x^2\): \(-6x^2 - 8x^2 - 2x^2 = -16x^2\)
- For \(x\): \(12x - 4x = 8x\)
- Constant term: \(6\) (no like terms)

So, the expanded expression in standard form is:

\[
2x^4 - 16x^2 + 8x + 6
\]

Would you like further details or have any questions? Here are five related questions:

1. How do you expand expressions with more than two terms in each polynomial?
2. What is the importance of combining like terms in polynomial expansion?
3. How would you factor the expanded polynomial back into its original form?
4. What is the degree of the polynomial \(2x^4 - 16x^2 + 8x + 6\)?
5. How does polynomial multiplication differ from polynomial addition?

**Tip:** When expanding polynomials, carefully track each term to ensure accuracy, particularly when dealing with negative signs.

Expand the expression to a polynomial in standard form: (x^2 - 2x - 1)(2x^2 + 4x - 6)

Explore the step-by-step expansion and simplification of the polynomial expression (x^2 - 2x - 1)(2x^2 + 4x - 6). This solution covers key algebraic concepts such as the distributive property and combining like terms. Ideal for students in Grades 9-11.

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