Question

What is the product?

x2_16_x® − 2x2 + x
12x48 x2+3x-4
O
O
x(x-4)(x-1)]
2(x+4)
x(x-1)
2
(x+4)(x-4)
|2x(x-1)|
(x-4)(x-1)
||2x(x+4)

Answer 1

The final product is:
`\[ (x(x-4)(x-1)))/(2(x + 4)) \]`

Step 1: Factor the numerators and denominators:

Numerator 1:

x^2 - 16 = (x+4)(x-4) (factor the difference of two squares)

Numerator 2:

x^2 - 2x^2 + x = x(x-1) (factor out x)

Denominator 1:

2x + 8 = 2(x+4) (factor out 2)

Denominator 2:

x^2 + 3x - 4 = (x+4)(x-1) (factor using the sum-product pattern)

Step 2: Combine the factors:

Numerator:

(x+4)(x-4) * x(x-1) = x(x+4)(x-4)(x-1)

Denominator:

2(x+4) * (x+4)(x-1) = 2(x+4)^2(x-1)

Step 3: Divide numerator and denominator:

x(x+4)(x-4)(x-1) / 2(x+4)^2(x-1)

Canceling common factors:

x(x-4)(x-1) / 2(x+4)(x-1)

Simplifying further:

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

Therefore, the verified product of the two expressions is:

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

Which is the same as your answer:

`\[ (x(x-4)(x-1)))/(2(x + 4)) \]`

Both forms are correct, though the first form might be considered slightly more concise.