To solve the equation:

\[
261x^3 + 309x^2 + 119x = -14
\]

First, let's rewrite the equation as:

\[
261x^3 + 309x^2 + 119x + 14 = 0
\]

Now, we'll solve this cubic equation using algebraic methods or numerical approximation. I'll start by attempting possible rational roots and then proceed from there.

Let me work this out.The equation \( 261x^3 + 309x^2 + 119x + 14 = 0 \) has complex solutions, which involve irrational and imaginary components. Therefore, finding exact numeric values for the roots would require further simplification or approximation.

Let me calculate approximate numerical solutions to make them more understandable.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can still help by explaining the steps to approximate the solutions manually or using another method.

Let me know if you'd like details on how to proceed, or feel free to try again later!

Here are some related questions to consider:
1. What are the general steps for solving cubic equations?
2. How can you approximate complex roots of a cubic equation?
3. What methods are used to find the rational roots of a polynomial?
4. How does the cubic formula work for solving higher-degree polynomials?
5. Can you use graphing tools to find approximate solutions for polynomial equations?

**Tip:** When solving cubic equations, trying rational root tests can help simplify before using advanced methods!

Learn how to solve the cubic equation 261x^3 + 309x^2 + 119x = -14. This detailed guide covers key algebraic concepts such as the Rational Root Theorem, numerical approximations, and complex roots. Suitable for students in Grades 10-12.

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