Add or subtract, as indicated. See Example 4. 2(12y² - 8y + 6) - 4(3y² - 4y +2)

Verified step by step guidance

1. Distribute the 2 across each term inside the first parentheses: multiply 2 by each term in \$12y^{2} - 8y + 6$, resulting in \(2 \times 12y^{2}\), \(2 \times (-8y)\), and \(2 \times 6\).
2. Distribute the -4 across each term inside the second parentheses: multiply -4 by each term in \$3y^{2} - 4y + 2$, resulting in \(-4 \times 3y^{2}\), \(-4 \times (-4y)\), and \(-4 \times 2\).
3. Write out the expanded expression after distribution, combining all the terms obtained from both distributions.
4. Group like terms together: combine the \(y^{2}\) terms, the \(y\) terms, and the constant terms separately.
5. Simplify each group by adding or subtracting the coefficients of like terms to get the final 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 is essential for expanding expressions like 2(12y² - 8y + 6) by multiplying 2 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, 12y² and 3y² are like terms and can be combined by adding or subtracting their coefficients. This simplifies the expression after distribution.

Polynomial Subtraction

Polynomial subtraction requires careful handling of the minus sign before parentheses. When subtracting polynomials, you must distribute the negative sign to each term inside the second parentheses before combining like terms. This ensures correct simplification of the expression.