To solve the cubic equation \( z^3 + 4z^2 + z + 1 = 0 \), we can follow these steps:

### Step 1: Check for Rational Roots
Using the Rational Root Theorem, the possible rational roots are \( \pm 1 \), because the constant term and the leading coefficient are both 1.

Let's test these roots:

- For \( z = 1 \):
  \[
  1^3 + 4(1)^2 + 1(1) + 1 = 1 + 4 + 1 + 1 = 7 \quad (\text{not a root})
  \]

- For

Learn how to solve the cubic equation z^3 + 4z^2 + z + 1 = 0 with detailed steps and explanations. This problem involves applying the Rational Root Theorem and testing potential roots. Suitable for advanced high school students.

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