factor 2x^4+12x^3-18x^2-108x
2x^4+12x^3-18x^2-108x
2x(x^3+6x^2-9x-54)
Now use grouping
2x(x^2(x+6)-9(x+6))
2x(x^2-9)(x+6)
2x(x-3)(x+3)(x+6)
To factor the expression 2x^4 + 12x^3 - 18x^2 - 108x, we can follow these steps:
Step 1: Look for the greatest common factor (GCF).
In this case, we can factor out a 2x from each term, as it is a common factor:
2x(x^3 + 6x^2 - 9x - 54)
Step 2: Group the terms and factor by grouping.
Next, we group the terms together:
2x[(x^3 + 6x^2) + (-9x - 54)]
Step 3: Factor out the GCF from each grouping.
Now, we can factor out the GCF from each group:
2x[x^2(x + 6) - 9(x + 6)]
Step 4: Factor out the common binomial factor.
We have a common binomial factor of (x + 6), so we can factor that out:
2x(x + 6)(x^2 - 9)
Step 5: Further factorize if possible.
The expression (x^2 - 9) is a difference of squares, which can be factorized as follows:
2x(x + 6)(x - 3)(x + 3)
Therefore, the fully factorized form of 2x^4 + 12x^3 - 18x^2 - 108x is 2x(x + 6)(x - 3)(x + 3).