Question

multiply.
$(2y^2 - 6y + 7)(2y^2 + 4y - 5)$
simplify your answer.

Explanation:

Step1: Apply distributive property (FOIL for polynomials)

Multiply each term in the first polynomial by each term in the second polynomial:
\[

$$\begin{align*} &(2y^2 - 6y + 7)(2y^2 + 4y - 5)\\ =& 2y^2(2y^2) + 2y^2(4y) + 2y^2(-5) - 6y(2y^2) - 6y(4y) - 6y(-5) + 7(2y^2) + 7(4y) + 7(-5) \end{align*}$$

\]

Step2: Simplify each product

Calculate each term:
\[

$$\begin{align*} &4y^4 + 8y^3 - 10y^2 - 12y^3 - 24y^2 + 30y + 14y^2 + 28y - 35 \end{align*}$$

\]

Step3: Combine like terms

Combine the terms with the same power of \(y\):

• For \(y^4\): \(4y^4\)
• For \(y^3\): \(8y^3 - 12y^3 = -4y^3\)
• For \(y^2\): \(-10y^2 - 24y^2 + 14y^2 = -20y^2\)
• For \(y\): \(30y + 28y = 58y\)
• Constant term: \(-35\)

So the simplified form is:
\[
4y^4 - 4y^3 - 20y^2 + 58y - 35
\]

Answer:

\(4y^4 - 4y^3 - 20y^2 + 58y - 35\)