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### Problem 12: 
Solve the equation \( \sqrt{2x + 1} + \sqrt[3]{x + 4} = 5 \) and find the value of \( x^2 \).

#### Steps to solve:
1. Isolate one of the terms, say \( \sqrt{2x + 1} \), by moving the other term to the other side:
   \[
   \sqrt{2x + 1} = 5 - \sqrt[3]{x + 4}
   \]
2. Square both sides to eliminate the square root:
   \[
   2x + 1 = (5 - \sqrt[3]{x + 4})^2
   \]
   This expansion requires careful simplification.

3. Further steps involve expanding and solving the resulting polynomial equation.

Solve the equation: √(2x + 1) + ³√(x + 4) = 5 and find the value of x². Additionally, find the value of 'm' given the two equations x³ + mx² + 18 = 0 and x³ + nx + 12 = 0, where they share two common roots.

This page provides a step-by-step solution for solving the radical equation √(2x + 1) + ³√(x + 4) = 5 and the cubic equations x³ + mx² + 18 = 0 and x³ + nx + 12 = 0 with two common roots. Concepts include solving radical equations, understanding cubic roots, and factoring polynomials. Suitable for advanced algebra students in Grades 10-12.

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