Perform the indicated operations. $-3(4q^2-3q+2)+2(-q^2+q-4)$

Verified step by step guidance

1. Identify the expression to simplify: \(-3(4q^2 - 3q + 2) + 2(-q^2 + q - 4)\).
2. Distribute the constants outside the parentheses to each term inside the parentheses: multiply \(-3\) by each term in \((4q^2 - 3q + 2)\) and multiply \(2\) by each term in \((-q^2 + q - 4)\).
3. Write the distributed terms explicitly: \(-3 \times 4q^2\), \(-3 \times (-3q)\), \(-3 \times 2\), and \(2 \times (-q^2)\), \(2 \times q\), \(2 \times (-4)\).
4. Simplify each multiplication to get the new terms: \(-12q^2\), \(+9q\), \(-6\), \(-2q^2\), \(+2q\), and \(-8\).
5. Combine like terms by grouping the \(q^2\) terms together, the \(q\) terms together, and the constant terms together to write the simplified expression.

Key Concepts

Distributive Property

The distributive property allows you to multiply a single term by each term inside a parenthesis. For example, a(b + c) = ab + ac. This property is essential for expanding expressions like -3(4q^2 - 3q + 2) by multiplying -3 with each term inside the parentheses.

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. For instance, 4q^2 and -2q^2 are like terms and can be combined to 2q^2. This step simplifies the expression after distribution.

Polynomial Operations

Polynomial operations include addition, subtraction, and multiplication of polynomial expressions. Understanding how to perform these operations correctly is crucial for simplifying expressions such as the given problem, which involves multiplying and then adding polynomials.