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Factoring the Expression (x^2 + x - 6)(2x^2 + 4x)
To factor the expression (x^2 + x - 6)(2x^2 + 4x), we need to factor each individual expression within the parentheses and then combine the factors.
Step 1: Factor (x^2 + x - 6)
This is a quadratic expression that can be factored by finding two numbers that add up to 1 (the coefficient of the x term) and multiply to -6 (the constant term). The numbers 3 and -2 satisfy these conditions. Therefore, we can factor (x^2 + x - 6) as:
(x + 3)(x - 2)
Step 2: Factor (2x^2 + 4x)
This expression has a common factor of 2x. We can factor out 2x to obtain:
2x(x + 2)
Step 3: Combine the Factors
Now that we have factored both expressions, we can combine them:
(x + 3)(x - 2) * 2x(x + 2)
Step 4: Simplify
We can rearrange the factors and multiply the constants to get the final factored form:
2x(x + 3)(x - 2)(x + 2)
Therefore, the fully factored form of the expression (x^2 + x - 6)(2x^2 + 4x) is 2x(x + 3)(x - 2)(x + 2).