To solve the cubic equation 

\[
0 = x^3 + 4x^2 - 36x - 144,
\]

we can use the **Rational Root Theorem** and polynomial factoring to find the roots.

### Step-by-Step Solution:

1. **List possible rational roots:**
   According to the Rational Root Theorem, possible rational roots are the factors of the constant term (-144) divided by the factors of the leading coefficient (1).

   The constant term is \(-144\), and the leading coefficient is \(1\).

   Factors of \(-144\) are: \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 9, \pm 12, \pm 16, \pm 18, \pm 24, \pm 36, \pm 48, \pm 72, \pm 144\).

   Since the leading coefficient is \(1\), the possible rational roots are the same as the factors of \(-144\).

2. **Test the possible roots:**
   We will use synthetic division or substitution to test which possible roots actually satisfy the equation.

   - Let's start by testing \(x = 4\).

   Substitute \(x = 4\) into the equation:

   \[
   4^3 + 4(4^2) - 36(4) - 144 = 64 + 64 - 144 - 144 = 0.
   \]

   Since \(x = 4\) satisfies the equation, it is a root of the cubic equation.

3. **Factor the cubic polynomial using the root:**
   Since \(x = 4\) is a root, we can factor the polynomial by \((x - 4)\).

   We can use **synthetic division** to divide \(x^3 + 4x^2 - 36x - 144\) by \(x - 4\).

   The coefficients of the polynomial are \(1, 4, -36, -144\).

   Perform synthetic division:

   ```
   4 |   1    4   -36  -144
     |        4    32   -16
     ---------------------
         1    8    -4   0
   ```

   The quotient is \(x^2 + 8x - 4\) and the remainder is \(0\).

   Therefore, the cubic polynomial can be factored as:

   \[
   x^3 + 4x^2 - 36x - 144 = (x - 4)(x^2 + 8x - 36).
   \]

4. **Solve the quadratic equation:**
   Next, we solve the quadratic equation \(x^2 + 8x - 36 = 0\) using the quadratic formula:

   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
   \]

   where \(a = 1\), \(b = 8\), and \(c = -36\).

   Calculate the discriminant:

   \[
   b^2 - 4ac = 8^2 - 4(1)(-36) = 64 + 144 = 208.
   \]

   Since the discriminant is positive, there are two real roots. Now find these roots:

   \[
   x = \frac{-8 \pm \sqrt{208}}{2}.
   \]

   Simplify \(\sqrt{208}\):

   \[
   \sqrt{208} = \sqrt{16 \times 13} = 4\sqrt{13}.
   \]

   So, the solutions are:

   \[
   x = \frac{-8 \pm 4\sqrt{13}}{2} = -4 \pm 2\sqrt{13}.
   \]

5. **List all roots:**
   The roots of the equation \(x^3 + 4x^2 - 36x - 144 = 0\) are:

   \[
   x = 4, \quad x = -4 + 2\sqrt{13}, \quad x = -4 - 2\sqrt{13}.
   \]

### Final Answer:

\[
x = 4, \quad x = -4 + 2\sqrt{13}, \quad x = -4 - 2\sqrt{13}.
\]

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

Here are 5 related questions you might find helpful:

1. How can you solve a quadratic equation using the completing square method?
2. What are the different methods to solve cubic equations?
3. How do you use synthetic division to factor polynomials?
4. What is the significance of the discriminant in quadratic equations?
5. How do you find complex roots of a polynomial equation?

**Tip:** When solving cubic equations, finding one real root can simplify the problem significantly, allowing you to factor the cubic polynomial and solve the remaining quadratic equation.

Learn how to solve the cubic equation x^3 + 4x^2 - 36x - 144 with a detailed step-by-step solution using the Rational Root Theorem and polynomial factoring. This problem covers key algebraic concepts, including cubic equations and quadratic equations. Suitable for students in Grades 10-12.

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