Perform the indicated operations. See Examples 2–6. -3(4q^2-3q+2) + 2(-q^2+q-4)

Verified step by step guidance

1. Distribute the -3 across the terms inside the first parentheses: \(-3(4q^2 - 3q + 2)\).
2. This results in: \(-3 \cdot 4q^2 + (-3) \cdot (-3q) + (-3) \cdot 2\).
3. Simplify each term: \(-12q^2 + 9q - 6\).
4. Next, distribute the 2 across the terms inside the second parentheses: \(2(-q^2 + q - 4)\).
5. This results in: \(2 \cdot (-q^2) + 2 \cdot q + 2 \cdot (-4)\), which simplifies to \(-2q^2 + 2q - 8\).

Key Concepts

Distributive Property

The distributive property states that a(b + c) = ab + ac. This property allows us to multiply a single term by each term within a parenthesis. In the given expression, we apply this property to distribute -3 and 2 across the polynomials, ensuring that each term is multiplied correctly.

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. After distributing the constants in the expression, we will group the resulting terms based on their variable components, simplifying the expression into a more manageable form.

Polynomial Operations

Polynomial operations include addition, subtraction, and multiplication of polynomial expressions. In this question, we are performing both distribution and combination of like terms, which are essential steps in simplifying polynomial expressions and solving algebraic equations.