We are tasked with solving the cubic equation:

$x^3 + 4x^2 + x + 4 = 0$

Step 1: Group terms and look for factors

Group terms to see if there are any common factors or patterns: $x^3 + 4x^2 + x + 4 = (x^3 + 4x^2) + (x + 4).$ Factor each group: $x^2(x + 4) + 1(x + 4).$ Now factor out $x + 4$: $(x + 4)(x^2 + 1) = 0.$

Step 2: Solve for $x$

Set each factor to 0:

1. $x + 4 = 0 \quad \Rightarrow \quad x = -4.$
2. $x^2 + 1 = 0 \quad \Rightarrow \quad x^2 = -1.$

Step 3: Analyze the solutions

• $x = -4$ is a real solution.
• $x^2 = -1$ does not yield any real solutions because the square root of a negative number is not real. The solutions to this part are complex: $x = \pm i$.

Final Answer:

The only real solution is: $x = -4.$

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Relative Questions:

1. How do you find complex solutions for cubic equations?
2. What is the significance of factoring in solving polynomial equations?
3. How do we confirm if a cubic equation has real or complex roots without solving it?
4. How does the Fundamental Theorem of Algebra apply here?
5. What are some numerical methods for solving cubic equations when factoring fails?

Tip: Always check for common factors or grouping when solving polynomial equations—it can simplify the process significantly!