Question

1. what is the quotient represented by the expression \\(\frac{5x^2 - 15x - 20}{5x + 5}\\) for all values of a where the expression is defined?

a. \\(x + 4\\)
b. \\(x - 4\\)
c. \\(2x - 4\\)
d. \\(2x + 4\\)

Explanation:

Step1: Factor the numerator

Factor $5x^2 -15x -20$:
First factor out 5: $5(x^2 - 3x - 4)$
Then factor the quadratic: $5(x-4)(x+1)$
So the numerator becomes $5(x-4)(x+1)$

Step2: Rewrite the original expression

Substitute factored numerator:
$\frac{5(x-4)(x+1)}{5x+5}$

Step3: Factor the denominator

Factor $5x+5$: $5(x+1)$

Step4: Cancel common factors

Cancel $5(x+1)$ (valid where $x
eq -1$, since expression is undefined at $x=-1$):
$\frac{\cancel{5(x+1)}(x-4)}{\cancel{5(x+1)}} = x-4$

Answer:

b. $x - 4$