https://file.aiquickdraw.com/mathbot/file/17272227957192vm5ef9s.jpg

The equation extracted from the image is:

\[
x^3 + x^2 - x - 1 = 0
\]

Let's factor the cubic equation to find the real solutions.

### Step 1: Try Factoring by Grouping
We can attempt to group terms:

\[
x^3 + x^2 - x - 1 = (x^2(x + 1)) - (1(x + 1))
\]

Now, factor out the common factor \((x + 1)\):

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

### Step 2: Factor Further
Notice that \(x^2 - 1\) is a difference of squares, so it can be factored as:

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

Thus, the full factorization of the original equation is:

\[
(x - 1)(x + 1)(x + 1) = 0
\]

### Step 3: Solve for \(x\)
Set each factor equal to 0:

1. \(x - 1 = 0 \quad \Rightarrow \quad x = 1\)
2. \(x + 1 = 0 \quad \Rightarrow \quad x = -1\)

Thus, the real solutions are \(x = 1\) and \(x = -1\) (with \(x = -1\) having a multiplicity of 2).

### Final Solutions:
\[
x = 1, \quad x = -1
\]

Would you like more details on any step, or have further questions?

Here are some related questions:
1. How do you handle higher-order polynomials that can't be factored easily?
2. What are the benefits of factoring over other methods like synthetic division?
3. Can factoring always find all solutions, or are there cases where another method is required?
4. How can you verify that your factorization is correct?
5. What do you do if a polynomial has complex roots?

**Tip:** Always check for special factorization patterns like difference of squares or perfect cubes, as they can simplify the process greatly.

Find the real solutions of the following equation by factoring: x^3 + x^2 - x - 1 = 0

Learn how to solve the cubic equation x^3 + x^2 - x - 1 = 0 by factoring. This solution includes key algebraic concepts such as factoring by grouping, difference of squares, and the zero product property. Suitable for students in Grades 9-12.

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