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The given equation is:

\[
2x^3 + 5x^2 - 14x - 8 = 0
\]

We are also told that one factor is \( x + 4 \). This implies \( x = -4 \) is a root of the equation. We can use synthetic or polynomial division to divide the cubic polynomial by \( x + 4 \) to simplify the equation and then solve for the remaining roots.

Let's proceed with polynomial division.After performing synthetic division, the cubic polynomial \( 2x^3 + 5x^2 - 14x - 8 \) is reduced to:

\[
2x^2 - 3x - 2 = 0
\]

This is a quadratic equation. Now, let's solve this quadratic to find the remaining roots.The quadratic equation \( 2x^2 - 3x - 2 = 0 \) has the following solutions:

\[
x = 2 \quad \text{and} \quad x = -0.5
\]

Thus, the full set of solutions to the original cubic equation \( 2x^3 + 5x^2 - 14x - 8 = 0 \) is:

\[
x = -4, \quad x = 2, \quad \text{and} \quad x = -0.5
\]

Would you like more details on any step? Here are some related questions:

1. How is synthetic division performed in detail?
2. Can you explain the quadratic formula in depth?
3. What are the properties of cubic equations?
4. How can we verify these roots?
5. What are the real-world applications of solving cubic equations?

**Tip:** Always check the discriminant \( b^2 - 4ac \) when solving quadratic equations to determine if the roots are real or complex.

Solve the equation: 2x^3 + 5x^2 - 14x - 8 = 0; one factor is x + 4.

This example solves the cubic equation 2x^3 + 5x^2 - 14x - 8 = 0 by factoring, using synthetic division and solving a quadratic equation. Learn how to approach complex polynomials step-by-step.

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