https://file.aiquickdraw.com/mathbot/file/1729780751683691d9axn.jpg

The equation you provided is:

\[
0 = (x^2 + 1)(2x + 4)(x + 2)
\]

To solve this equation, let's break it down step by step.

### Step 1: Analyze each factor

- The equation is already factored into three terms:
  1. \(x^2 + 1\)
  2. \(2x + 4\)
  3. \(x + 2\)

The equation equals zero, so by the **Zero Product Property**, at least one of the factors must equal zero.

### Step 2: Solve \(x^2 + 1 = 0\)
- \(x^2 + 1 = 0\) 
  \[
  x^2 = -1
  \]
  This has no real solution, but it has **complex solutions**:
  \[
  x = \pm i
  \]
  where \(i\) is the imaginary unit.

### Step 3: Solve \(2x + 4 = 0\)
- \(2x + 4 = 0\)
  \[
  2x = -4
  \]
  \[
  x = -2
  \]

### Step 4: Solve \(x + 2 = 0\)
- \(x + 2 = 0\)
  \[
  x = -2
  \]

### Conclusion:
- The real solution to this equation is \(x = -2\), which comes from both \(2x + 4 = 0\) and \(x + 2 = 0\).
- The complex solutions are \(x = i\) and \(x = -i\).

Do you need further details or explanations? Here are some related questions to explore:

1. How do you apply the Zero Product Property to solve factored equations?
2. What is the significance of complex numbers in this solution?
3. Can you explain why \(x^2 + 1 = 0\) has no real solutions?
4. How can you simplify expressions like \(2x + 4\) to find the roots?
5. What other types of equations involve complex numbers as solutions?

**Tip:** Always check if a quadratic factor yields real or complex roots, as it affects the nature of your solutions.

Solve the equation 0 = (x^2 + 1)(2x + 4)(x + 2)

Learn how to solve the factored equation 0 = (x^2 + 1)(2x + 4)(x + 2). This problem explores algebraic concepts such as factoring, the Zero Product Property, and complex number solutions. Suitable for Grades 10-12.

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