To solve the equation [tex]\((x + 1)(x^2 + x + 1) - (x - 1)(x^2 + x + 1) = 2(x - 2)\)[/tex], we need to follow these steps:
1. Expand both sides:
[tex]\((x + 1)(x^2 + x + 1) - (x - 1)(x^2 + x + 1) = 2(x - 2)\)[/tex]
2. First, expand the left-hand side:
[tex]\((x + 1)(x^2 + x + 1)\)[/tex] and [tex]\((x - 1)(x^2 + x + 1)\)[/tex]
Let's expand [tex]\((x + 1)(x^2 + x + 1)\)[/tex]:
[tex]\[
(x + 1)(x^2 + x + 1) = x(x^2 + x + 1) + 1(x^2 + x + 1) = x^3 + x^2 + x + x^2 + x + 1 = x^3 + 2x^2 + 2x + 1
\][/tex]
Now expand [tex]\((x - 1)(x^2 + x + 1)\)[/tex]:
[tex]\[
(x - 1)(x^2 + x + 1) = x(x^2 + x + 1) - 1(x^2 + x + 1) = x^3 + x^2 + x - x^2 - x - 1 = x^3 - 1
\][/tex]
3. Combine the expanded forms:
Now substitute these back into the equation:
[tex]\[
x^3 + 2x^2 + 2x + 1 - (x^3 - 1) = 2(x - 2)
\][/tex]
Simplify the left-hand side:
[tex]\[
(x^3 + 2x^2 + 2x + 1) - x^3 + 1 = 2(x - 2)
\][/tex]
Combine like terms on the left:
[tex]\[
x^3 + 2x^2 + 2x + 1 - x^3 + 1 = 2(x - 2)
\][/tex]
[tex]\[
2x^2 + 2x + 2 = 2(x - 2)
\][/tex]
4. Simplify further:
We notice that [tex]\(2\)[/tex] is a common factor on the left side:
[tex]\[
2(x^2 + x + 1) = 2(x - 2)
\][/tex]
Divide both sides by 2:
[tex]\[
x^2 + x + 1 = x - 2
\][/tex]
5. Move all terms to one side to form a standard quadratic equation:
[tex]\[
x^2 + x + 1 - x + 2 = 0
\][/tex]
Simplify:
[tex]\[
x^2 + 3 = 0
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[
x^2 + 3 = 0
\][/tex]
Subtract 3 from both sides:
[tex]\[
x^2 = -3
\][/tex]
Take the square root of both sides:
[tex]\[
x = \pm \sqrt{-3}
\][/tex]
Since [tex]\(\sqrt{-3} = \sqrt{3}i\)[/tex], we have:
[tex]\[
x = \pm \sqrt{3}i
\][/tex]
So, the solution to the equation [tex]\((x + 1)(x^2 + x + 1) - (x - 1)(x^2 + x + 1) = 2(x - 2)\)[/tex] is:
[tex]\[
x = \pm \sqrt{3}i
\][/tex]
Or more explicitly:
[tex]\[
x = -\sqrt{3}i \quad \text{and} \quad x = \sqrt{3}i
\][/tex]
Thus, the solutions are [tex]\(x = -\sqrt{3}i\)[/tex] and [tex]\(x = \sqrt{3}i\)[/tex].
