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Factoring and Simplifying the Expression (x^2 + x - 6)(2x^2 + 4x)
This expression represents the product of two quadratic polynomials. To simplify it, we can use the distributive property and then factor the resulting polynomial.
Step 1: Distribute
We start by expanding the product:
(x^2 + x - 6)(2x^2 + 4x) = x^2(2x^2 + 4x) + x(2x^2 + 4x) - 6(2x^2 + 4x)
This simplifies to:
2x^4 + 4x^3 + 2x^3 + 4x^2 - 12x^2 - 24x
Step 2: Combine like terms
Combining the terms with the same powers of x:
2x^4 + 6x^3 - 8x^2 - 24x
Step 3: Factor out common factors
We can factor out a 2x from each term:
2x(x^3 + 3x^2 - 4x - 12)
Step 4: Factor the cubic polynomial
The cubic polynomial inside the parentheses can be factored by grouping:
2x(x^3 + 3x^2 - 4x - 12) = 2x[x^2(x + 3) - 4(x + 3)]
Finally, we can factor out (x+3):
2x(x^2(x + 3) - 4(x + 3)) = 2x(x + 3)(x^2 - 4)
Step 5: Factor the difference of squares
The term (x^2 - 4) is a difference of squares and can be factored as:
2x(x + 3)(x^2 - 4) = 2x(x + 3)(x + 2)(x - 2)
Conclusion
Therefore, the simplified and fully factored form of (x^2 + x - 6)(2x^2 + 4x) is 2x(x + 3)(x + 2)(x - 2).