product of the polynomials p(x)=2x2+2x+2 and q(x)=x2−x−1

To find the product of the polynomials \( p(x) = 2x^2 + 2x + 2 \) and \( q(x) = x^2 - x - 1 \), we'll multiply them together step-by-step. Here's how you can do it:

1. Write down the polynomials:
   \[
   p(x) = 2x^2 + 2x + 2
   \]
   \[
   q(x) = x^2 - x - 1
   \]

2. Multiply each term in \( p(x) \) by each term in \( q(x) \):

   \[
   (2x^2 + 2x + 2) \cdot (x^2 - x - 1)
   \]

3. Distribute each term of \( p(x) \):

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

4. Compute each multiplication:

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

5. Combine like terms:

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

   \[
   = 2x^4 + 0x^3 - 2x^2 - 4x - 2
   \]

6. Simplify the expression:

   \[
   = 2x^4 - 2x^2 - 4x - 2
   \]

So, the product of the polynomials \( p(x) = 2x^2 + 2x + 2 \) and \( q(x) = x^2 - x - 1 \) is:

\[
2x^4 - 2x^2 - 4x - 2
\]

Would you like more details or have any questions?

Here are some questions you might ask:
1. How do you multiply two binomials?
2. What are like terms in polynomials?
3. How do you factor a polynomial?
4. What is the degree of a polynomial?
5. How do you find the roots of a polynomial?

**Tip:** Always combine like terms carefully after distributing when multiplying polynomials to ensure accuracy.

Learn how to find the product of the polynomials 2x^2 + 2x + 2 and x^2 - x - 1 with a detailed step-by-step solution. This problem covers key algebraic concepts, including polynomials, multiplication of polynomials, and combining like terms. Suitable for students in Grades 9-12.

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