Find each product : $(3x-2)(4x^2+3x-5)$

Verified step by step guidance

1. Identify the two polynomials to be multiplied: \((3x - 2)\) and \((4x^2 + 3x - 5)\).
2. Apply the distributive property (also known as the FOIL method for binomials) by multiplying each term in the first polynomial by each term in the second polynomial.
3. Multiply \$3x$ by each term in the second polynomial: \(3x \times 4x^2\), \(3x \times 3x\), and \(3x \times (-5)\).
4. Multiply \(-2\) by each term in the second polynomial: \(-2 \times 4x^2\), \(-2 \times 3x\), and \(-2 \times (-5)\).
5. Combine all the products from the previous steps and then simplify by combining like terms to write the final expanded polynomial.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in one polynomial by every term in the other polynomial. This process requires distributing each term carefully to ensure all products are accounted for, combining like terms afterward to simplify the expression.

Distributive Property

The distributive property states that a(b + c) = ab + ac. It is essential for multiplying polynomials because it allows you to multiply each term inside the parentheses by the term outside, ensuring all parts of the expression are multiplied correctly.

Combining Like Terms

After multiplying polynomials, you often get terms with the same variable raised to the same power. Combining like terms means adding or subtracting these terms to simplify the polynomial into its standard form, making it easier to interpret and use.