Final answer:
The product of prime polynomials equivalent to 8x⁴ + 36x³ - 72x² after factoring out the greatest common factor and using factoring by grouping is 8x²(x + 3)(x - 3).
Step-by-step explanation:
The question asks which product of prime polynomials is equivalent to 8x⁴ + 36x³ - 72x². To solve this, we factor by grouping or look for common factors. First, we can factor out the greatest common factor, which is 4x², getting 4x²(2x² + 9x - 18). Next, we factor the quadratic polynomial. Since 2 * (-18) = -36 and 9 is the middle term, we look for two numbers that multiply to -36 and add up to 9. These numbers are 12 and -3. So, we rewrite the quadratic as 4x²(2x² + 12x - 3x - 18) and factor by grouping to get 4x²(x(2x + 12) - 3(2x + 12)). Now we can factor out the common factor (2x + 12), leading us to 4x²(2x + 12)(x - 3). The polynomial 2x + 12 is not prime, so we can factor out a 2 to get 2(2x + 6), further simplifying our expression to 8x²(x + 3)(x - 3).
Now, let's factorize the quadratic expression inside the parentheses. We need to find two numbers that multiply to -9 and add up to 4. The numbers are 3 and -3:
8x⁴ + 36x³ - 72x² = 8x²(x + 3)(x - 3)
Therefore, the product of prime polynomials equivalent to 8x⁴ + 36x³ - 72x² is 8x²(x + 3)(x - 3).