Final answer:
To find the product of the two polynomials, we must use the distributive property to multiply each term in the first polynomial by each term in the second, then combine like terms. The answer to the correct process is (-14x⁴ + 71x³ - 57x² + 12x - 6), which does not match any of the provided options; hence there may be a mistake in the options given.
Step-by-step explanation:
The question asks to multiply two polynomials: ((-2x² + 9x - 3) * (7x² - 4x + 2)). Multiplication of polynomials involves applying the distributive property, also known as the FOIL method (First, Outer, Inner, Last terms).
Here's the step-by-step multiplication:
- First: (-2x²) * (7x²) = -14x⁴
- Outer: (-2x²) * (-4x) = 8x³
- Inner: (9x) * (7x²) = 63x³
- Last: (9x) * (-4x) = -36x²
Now we multiply the last term (-3) with each term of the second polynomial:
- (-3) * (7x²) = -21x²
- (-3) * (-4x) = 12x
- (-3) * (2) = -6
Adding all results together and combining like-terms:
- -14x⁴ + (8x³ + 63x³) + (-36x² - 21x²) + 12x - 6
- -14x⁴ + 71x³ - 57x² + 12x - 6
The correct expanded form of the polynomial is therefore (-14x⁴ + 71x³ - 57x² + 12x - 6). However, this option is not presented in the question's choices, indicating either the options are incorrect or there has been a typo in the provided options. The process used to reach this answer employs the concept of polynomials multiplication and combining like-terms.