Question

what is the product?\\((y^{2}+3y + 7)(8y^{2}+y + 1)\\)\\(\bigcirc\\ 8y^{4}+24y^{3}+60y^{2}+10y + 7\\)\\(\bigcirc\\ 8y^{4}+25y^{3}+4y^{2}+10y + 7\\)\\(\bigcirc\\ 8y^{4}+25y^{3}+60y^{2}+7y + 7\\)\\(\bigcirc\\ 8y^{4}+25y^{3}+60y^{2}+10y + 7\\)

Explanation:

Step1: Distribute \( y^2 \)

Multiply \( y^2 \) with each term in \( 8y^2 + y + 1 \):
\( y^2 \cdot 8y^2 = 8y^4 \), \( y^2 \cdot y = y^3 \), \( y^2 \cdot 1 = y^2 \).

Step2: Distribute \( 3y \)

Multiply \( 3y \) with each term in \( 8y^2 + y + 1 \):
\( 3y \cdot 8y^2 = 24y^3 \), \( 3y \cdot y = 3y^2 \), \( 3y \cdot 1 = 3y \).

Step3: Distribute \( 7 \)

Multiply \( 7 \) with each term in \( 8y^2 + y + 1 \):
\( 7 \cdot 8y^2 = 56y^2 \), \( 7 \cdot y = 7y \), \( 7 \cdot 1 = 7 \).

Step4: Combine like terms

• \( y^4 \): \( 8y^4 \)
• \( y^3 \): \( y^3 + 24y^3 = 25y^3 \)
• \( y^2 \): \( y^2 + 3y^2 + 56y^2 = 60y^2 \)
• \( y \): \( 3y + 7y = 10y \)
• Constant: \( 7 \)

Combine all terms: \( 8y^4 + 25y^3 + 60y^2 + 10y + 7 \).

Answer:

\( 8y^4 + 25y^3 + 60y^2 + 10y + 7 \) (the fourth option: \( 8y^4 + 25y^3 + 60y^2 + 10y + 7 \))