Math Problem Statement

\left(x+2\right)\left(2x^{2}+3x-2\right)

Solution

To expand the expression (x+2)(2x2+3x2)\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:

\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.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Binomial Expansion

Formulas

FOIL Method

Theorems

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Suitable Grade Level

Grades 9-12