To solve the equation \( x(2x + 3) = x^3 + 1 \), follow these steps:

1. **Expand the left-hand side**:
   \[
   x(2x + 3) = 2x^2 + 3x
   \]
   So the equation becomes:
   \[
   2x^2 + 3x = x^3 + 1
   \]

2. **Rearrange all terms to one side**:
   Subtract \( x^3 + 1 \) from both sides:
   \[
   2x^2 + 3x - x^3 - 1 = 0
   \]
   This simplifies to:
   \[
   -x^3 + 2x^2 + 3x - 1 = 0
   \]
   Or:
   \[
   x^3 - 2x^2 - 3x + 1 = 0
   \]

3. **Solve the cubic equation**:
   The equation is now \( x^3 - 2x^2 - 3x + 1 = 0 \). We can try to factor this cubic equation or use methods like synthetic division to find the roots. Let's check for possible rational roots using the Rational Root Theorem.

   Possible rational roots are \( \pm 1 \). Testing \( x = 1 \):
   \[
   (1)^3 - 2(1)^2 - 3(1) + 1 = 1 - 2 - 3 + 1 = -3 \neq 0
   \]
   Testing \( x = -1 \):
   \[
   (-1)^3 - 2(-1)^2 - 3(-1) + 1 = -1 - 2 + 3 + 1 = 1 \neq 0
   \]
   None of the possible simple roots work, so we'd use more advanced methods like numerical approximations or factoring techniques to find the real roots.

Would you like details on factoring this cubic, or perhaps applying numerical methods like the Newton-Raphson method?

### Follow-up questions:
1. How would you factor a cubic equation if a root was known?
2. What are some methods for solving higher-degree polynomial equations?
3. How does the Rational Root Theorem help in solving polynomial equations?
4. Can you explain synthetic division in detail?
5. What is the Newton-Raphson method and how does it work?

**Tip:** For cubic equations, checking for simple rational roots early can often simplify the problem before using more advanced techniques.

Learn how to solve the cubic equation x(2x + 3) = x^3 + 1 with a detailed step-by-step solution. This problem covers advanced algebraic concepts, including polynomial equations and the Rational Root Theorem. Suitable for students in Grades 10-12.

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