To expand the expression \(\left(x + 2\right)\left(2x^{2} + 3x - 2\right)\), you can apply the distributive property, often referred to as the FOIL method for binomials.

Here’s the step-by-step process:

\[
\begin{aligned}
\left(x + 2\right)\left(2x^{2} + 3x - 2\right) &= x \cdot \left(2x^{2} + 3x - 2\right) + 2 \cdot \left(2x^{2} + 3x - 2\right) \\
&= \left(x \cdot 2x^{2}\right) + \left(x \cdot 3x\right) + \left(x \cdot -2\right) + \left(2 \cdot 2x^{2}\right) + \left(2 \cdot 3x\right) + \left(2 \cdot -2\right) \\
&= 2x^{3} + 3x^{2} - 2x + 4x^{2} + 6x - 4.
\end{aligned}
\]

Next, combine like terms:

\[
2x^{3} + \left(3x^{2} + 4x^{2}\right) + \left(-2x + 6x\right) - 4 = 2x^{3} + 7x^{2} + 4x - 4.
\]

So, the expanded expression is:

\[
\boxed{2x^{3} + 7x^{2} + 4x - 4}
\]

Would you like further details or have any questions about this process?

Here are 5 related questions you might find interesting:

1. How do you factor the polynomial \(2x^3 + 7x^2 + 4x - 4\)?
2. What happens if you multiply \(\left(x + 2\right)\) by a quadratic expression instead of a cubic one?
3. How do you solve the equation \(2x^3 + 7x^2 + 4x - 4 = 0\)?
4. What are the roots of the polynomial \(2x^3 + 7x^2 + 4x - 4\)?
5. How do you find the derivative of the polynomial \(2x^3 + 7x^2 + 4x - 4\)?

**Tip:** When expanding products of polynomials, always combine like terms to simplify the expression.

Learn how to expand the expression (x + 2)(2x^2 + 3x - 2) using the distributive property (FOIL method). Step-by-step solution with combined like terms. Suitable for high school students studying algebra and polynomials.

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