Question

Expand the expression to a polynomial in standard form:
left parenthesis, 3, x, squared, minus, 2, x, minus, 2, right parenthesis, left parenthesis, 3, x, squared, plus, 2, x, plus, 4, right parenthesis
(3x
2
−2x−2)(3x
2
+2x+4)







Solve the problem just like the example below
⬇️
Expand the expression to a polynomial in standard form:
left parenthesis, x, squared, minus, x, minus, 2, right parenthesis, left parenthesis, minus, x, squared, plus, 3, x, minus, 6, right parenthesis
(x
2
−x−2)(−x
2
+3x−6)

minus, x, squared
−x
2
plus, 3, x
+3x
minus, 6
−6
x, squared
x
2
minus, x, to the power 4
−x
4
3, x, cubed
3x
3
minus, 6, x, squared
−6x
2
minus, x
−x
x, cubed
x
3
minus, 3, x, squared
−3x
2
6, x
6x
minus, 2
−2
2, x, squared
2x
2
minus, 6, x
−6x
12
12
Combine like terms:
Combine like terms:
minus, x, to the power 4 , plus, 4, x, cubed, minus, 7, x, squared, plus, 12
−x
4
+4x
3
−7x
2
+12

Answer 1

To expand the expression \((3x^2 - 2x - 2)(3x^2 + 2x + 4)\) and collect like terms, we can use the distributive property (also known as the FOIL method for binomials). Here's how we can break it down step by step:

1. **Expand Each Term:**
- Distribute each term in the first polynomial \((3x^2 - 2x - 2)\) to each term in the second polynomial \((3x^2 + 2x + 4)\).

First, multiply \(3x^2\) by each term:
- \(3x^2 \cdot 3x^2 = 9x^4\)
- \(3x^2 \cdot 2x = 6x^3\)
- \(3x^2 \cdot 4 = 12x^2\)

Next, multiply \(-2x\) by each term:
- \(-2x \cdot 3x^2 = -6x^3\)
- \(-2x \cdot 2x = -4x^2\)
- \(-2x \cdot 4 = -8x\)

Finally, multiply \(-2\) by each term:
- \(-2 \cdot 3x^2 = -6x^2\)
- \(-2 \cdot 2x = -4x\)
- \(-2 \cdot 4 = -8\)

2. **Combine all the terms together:**
\[
9x^4 + 6x^3 + 12x^2 - 6x^3 - 4x^2 - 8x - 6x^2 - 4x - 8
\]

3. **Combine like terms:**
- \(9x^4\) (only one term of \(x^4\))
- \(6x^3 - 6x^3 = 0\) (the \(x^3\) terms cancel out)
- \(12x^2 - 4x^2 - 6x^2 = 2x^2\)
- \(-8x - 4x = -12x\)
- \(-8\) (only one constant term)

Final result:
\[
9x^4 + 2x^2 - 12x - 8
\]

Thus, the expression in standard form is:
\[
9x^4 + 2x^2 - 12x - 8
\]