Factoring (x² + x  6)(2x² + 4x)
This problem involves factoring a product of two polynomials. Here's how to approach it:
Step 1: Factor each individual polynomial

x² + x  6: This is a quadratic trinomial. We need to find two numbers that add up to 1 (the coefficient of the x term) and multiply to 6 (the constant term). These numbers are 3 and 2. Therefore: x² + x  6 = (x + 3)(x  2)

2x² + 4x: This expression has a common factor of 2x. Factoring it out, we get: 2x² + 4x = 2x(x + 2)
Step 2: Combine the factored expressions
Now that we've factored each polynomial, we can substitute them back into the original expression:
(x² + x  6)(2x² + 4x) = (x + 3)(x  2) * 2x(x + 2)
Step 3: Simplify (optional)
While this is the completely factored form, we can simplify it further by rearranging the terms:
(x + 3)(x  2) * 2x(x + 2) = 2x(x + 3)(x  2)(x + 2)
Final Answer
The fully factored form of (x² + x  6)(2x² + 4x) is 2x(x + 3)(x  2)(x + 2).