Factor completely 4x3 + 8x2 − 25x − 50.
(4x2 − 25)(x + 2)
(2x − 2)(2x + 2)(x + 5)
(2x − 5)(2x + 5)(x + 2)
(2x − 5)(2x + 5)(x − 2)
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4x^3 + 8x^2 - 25x - 50
factor by grouping.
4x^2 (x+2) - 25(x+2)
(4x^2-25)(x+2)
(2x-5)(2x+5)(x+2)
To factor the expression 4x^3 + 8x^2 - 25x - 50 completely, we need to find the common factors among the terms and group them together.
First, we can notice that all the terms have a common factor of 2. So let's factor out a 2:
2(2x^3 + 4x^2 - 12.5x - 25)
Now, let's focus on the first two terms and the last two terms separately and factor them if possible.
The first two terms, 2x^3 + 4x^2, have a common factor of 2x^2. So we can factor it out:
2x^2(x + 2)
The last two terms, -12.5x - 25, do not have a common factor other than 1. However, they share a negative sign. So, we can factor out -1 to make them:
-1(12.5x + 25)
Now, let's combine the factored terms:
2x^2(x + 2) - 1(12.5x + 25)
Simplifying further, we get:
2x^2(x + 2) - 12.5(x + 2)
Now, notice that (x + 2) appears common in both terms. We can factor it out:
(x + 2)(2x^2 - 12.5)
Finally, the expression 4x^3 + 8x^2 - 25x - 50 is completely factored as (x + 2)(2x^2 - 12.5).