Question

\\\frac{-2x^2 + 2}{x^2 - x - 20} \div \frac{2x^2 - x - 3}{x^2 + 8x + 16} \cdot \frac{6x^2 - 7x - 3}{12x^2 - 6x}\\

question 8 (4 points)
is the following expression equivalent to the function in the previous question? explain and show your work.
\\\frac{-3x^3 - 10x^2 + 9x + 4}{6x^3 - 33x^2 + 15x}\\

Explanation:

Simplify the first rational expression and convert division to multiplication

$$
LATEXBLOCK0
$$

Cancel common factors to find the simplified form of the first expression

$$
LATEXBLOCK1
$$

Compare the simplified expression with the given expression

$$
LATEXBLOCK2
$$

Answer:

Yes, the expression is equivalent to the function in the previous question.

By factoring and simplifying the first expression:

$$ \frac{-2(x-1)(x+1)}{(x-5)(x+4)} \cdot \frac{(x+4)^2}{(2x-3)(x+1)} \cdot \frac{(2x-3)(3x+1)}{6x(2x-1)} = \frac{-(x-1)(x+4)(3x+1)}{3x(x-5)(2x-1)} $$

Expanding the simplified numerator and denominator:

$$ \frac{-(3x^3 + 10x^2 - 9x - 4)}{3x(2x^2 - 11x + 5)} = \frac{-3x^3 - 10x^2 + 9x + 4}{6x^3 - 33x^2 + 15x} $$

This matches the given expression exactly.