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------------------------------------------------ Which expression can be used to find the product of the polynomials \((9 - 3x^2)\) and \((-8x^2 + 4x + 5)\)?

A) \(27x^4 - 27x^2 - 45\)

B) \(-72x^4 + 36x^3 - 45x^2\)

C) \(72x^4 - 36x^3 + 45x^2\)

D) \(-27x^4 + 27x^2 + 45\)

To find the sum of the polynomials (9 – 3x^2) and (–8x^2 + 4x + 5) through multiplication, each term of the first polynomial is multiplied by each term of the second followed by combining like terms. The correct expression representing the sum is –72x^4 + 36x^3 – 45x^2, which is Option B.

Explanation:

To find the sum of the two given polynomials, (9 – 3x^2) and (–8x^2 + 4x + 5), we need to multiply them together. This process involves applying the distributive property, which entails multiplying each term in the first polynomial by each term in the second polynomial.

Let's do the multiplication step by step:

Multiply 9 by each term in the second polynomial: (9)(–8x^2) = –72x^2, (9)(4x) = 36x, and (9)(5) = 45.

Multiply – 3x^2 by each term in the second polynomial: (–3x^2)(–8x^2) = 24x^4, (–3x^2)(4x) = –12x^3, and (–3x^2)(5) = –15x^2.

Add the resulting terms together to get the final expression: –72x^4 + 24x^4 + 36x^3 – 12x^3 + 36x – 15x^2.

Combine like terms: –72x^4 + 24x^4 = –48x^4, 36x^3 – 12x^3 = 24x^3, and there are no other like terms to combine.

The final expression, which represents the sum of the two polynomials multiplied together, is –72x^4 + 36x^3 – 45x^2.

Thus, the correct answer is Option B: –72x^4 + 36x^3 – 45x^2.