Find each product. (m-n+k)(m+2n-3k)

Verified step by step guidance

1. Identify the two binomials to be multiplied: \((m - n + k)\) and \((m + 2n - 3k)\).
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 each term step-by-step: - \(m \times m\), - \(m \times 2n\), - \(m \times (-3k)\), - \((-n) \times m\), - \((-n) \times 2n\), - \((-n) \times (-3k)\), - \(k \times m\), - \(k \times 2n\), - \(k \times (-3k)\).
4. Write down all the products obtained from the previous step as a sum: \(m^2 + 2mn - 3mk - nm - 2n^2 + 3nk + km + 2kn - 3k^2\).
5. Combine like terms by grouping terms with the same variables and powers to simplify the expression.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in one polynomial by every term in the other polynomial. This process requires applying the distributive property to combine like terms and simplify the expression.

Distributive Property

The distributive property states that a(b + c) = ab + ac. It allows you to multiply a single term by each term inside a parenthesis, which is essential when expanding products of polynomials.

Combining Like Terms

After multiplying polynomials, terms with the same variables and exponents must be combined to simplify the expression. This step ensures the final product is written in its simplest form.