Question

1. $(6n^2 - 6n - 5)(7n^2 + 6n - 5)$

Explanation:

Step1: Apply distributive property (FOIL extended)

Multiply each term in the first polynomial by each term in the second polynomial:
$$(6n^2)(7n^2) + (6n^2)(6n) + (6n^2)(-5) + (-6n)(7n^2) + (-6n)(6n) + (-6n)(-5) + (-5)(7n^2) + (-5)(6n) + (-5)(-5)$$

Step2: Simplify each product

Calculate each term:

• \( (6n^2)(7n^2) = 42n^4 \)
• \( (6n^2)(6n) = 36n^3 \)
• \( (6n^2)(-5) = -30n^2 \)
• \( (-6n)(7n^2) = -42n^3 \)
• \( (-6n)(6n) = -36n^2 \)
• \( (-6n)(-5) = 30n \)
• \( (-5)(7n^2) = -35n^2 \)
• \( (-5)(6n) = -30n \)
• \( (-5)(-5) = 25 \)

Step3: Combine like terms

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

• \( n^4 \): \( 42n^4 \)
• \( n^3 \): \( 36n^3 - 42n^3 = -6n^3 \)
• \( n^2 \): \( -30n^2 - 36n^2 - 35n^2 = -101n^2 \)
• \( n \): \( 30n - 30n = 0n \) (this term cancels out)
• Constant: \( 25 \)

Answer:

\( 42n^4 - 6n^3 - 101n^2 + 25 \)