Question

Simplify completely quantity 2 x squared plus 20 x plus 32 all over x squared minus 2 x minus 80.

Group of answer choices
1 - negative 1 over 5
2 - negative 2 over 5
3 - 5 x over quantity x minus 5
4 - 2 open parentheses x plus 2 close parentheses over x minus 10

Answer 1

To simplify the expression, we factor both the numerator and denominator and then cancel out the common term, resulting in the simplified expression 2(x + 4) / (x - 10).

Step-by-step explanation:

To simplify the quantity 2x² + 20x + 32 over x² - 2x - 80, we need to factor both the numerator and the denominator. First, let's factor the numerator:

1. Looking for factors of 2x² + 20x + 32 that multiply to give us the constant term (32) times the coefficient of x² (2), which is 64, and add up to the coefficient of x (20).
2. We find that 8 and 8 are the numbers that satisfy these conditions.
3. Thus, we can write the numerator as (2x + 8)(x + 4).

Now, let's factor the denominator:

1. Similarly, we need factors of -80 that add up to the coefficient of x (-2).
2. The numbers -10 and 8 satisfy these conditions.
3. Therefore, the denominator can be factored as (x - 10)(x + 8).

Now we have the factored form of the quantity:

(2x + 8)(x + 4) / (x - 10)(x + 8)

We can see that (x + 8) is a common factor in both the numerator and the denominator, so we can cancel it out:

(2x + 8) / (x - 10)

Thus, the simplified form of the given quantity is:

2(x + 4) / (x - 10)