Question

Expand the expression to a polynomial in standard form:
left parenthesis, 2, x, squared, minus, x, plus, 3, right parenthesis, left parenthesis, x, squared, minus, 6, x, plus, 2, right parenthesis
(2x
2
−x+3)(x
2
−6x+2)






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

Let's expand both expressions step by step to achieve the polynomial in standard form.

### First Expression:
\[
(2x^2 - x + 3)(x^2 - 6x + 2)
\]

1. Distribute \(2x^2\):
\[
2x^2 \cdot (x^2 - 6x + 2) = 2x^4 - 12x^3 + 4x^2
\]

2. Distribute \(-x\):
\[
-x \cdot (x^2 - 6x + 2) = -x^3 + 6x^2 - 2x
\]

3. Distribute \(3\):
\[
3 \cdot (x^2 - 6x + 2) = 3x^2 - 18x + 6
\]

Now, combine all terms:
\[
(2x^4 - 12x^3 + 4x^2) + (-x^3 + 6x^2 - 2x) + (3x^2 - 18x + 6)
\]

Combine like terms:
- \(2x^4\)
- \((-12x^3 - x^3) = -13x^3\)
- \((4x^2 + 6x^2 + 3x^2) = 13x^2\)
- \((-2x - 18x) = -20x\)
- Constant term: \(6\)

Final expression in standard form:
\[
2x^4 - 13x^3 + 13x^2 - 20x + 6
\]

### Second Expression:
\[
(x^2 - x - 2)(-x^2 + 3x - 6)
\]

1. Distribute \(x^2\):
\[
x^2 \cdot (-x^2 + 3x - 6) = -x^4 + 3x^3 - 6x^2
\]

2. Distribute \(-x\):
\[
-x \cdot (-x^2 + 3x - 6) = x^3 - 3x^2 + 6x
\]

3. Distribute \(-2\):
\[
-2 \cdot (-x^2 + 3x - 6) = 2x^2 - 6x + 12
\]

Now, combine all terms:
\[
(-x^4 + 3x^3 - 6x^2) + (x^3 - 3x^2 + 6x) + (2x^2 - 6x + 12)
\]

Combine like terms:
- \(-x^4\)
- \((3x^3 + x^3) = 4x^3\)
- \((-6x^2 - 3x^2 + 2x^2) = -7x^2\)
- \((6x - 6x) = 0x\)
- Constant term: \(12\)

Final expression in standard form:
\[
-x^4 + 4x^3 - 7x^2 + 12
\]

### Summary of Results:
1. First Expression:
\[
2x^4 - 13x^3 + 13x^2 - 20x + 6
\]

2. Second Expression:
\[
-x^4 + 4x^3 - 7x^2 + 12
\]