(x^2+x-6)(2x^2+4x)=

2 min read Jun 17, 2024
(x^2+x-6)(2x^2+4x)=

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).

Related Post