(x^2+x)^2+4(x^2+x)-12

3 min read Jun 17, 2024

Factoring the Expression (x^2 + x)^2 + 4(x^2 + x) - 12

This article explores how to factor the expression (x^2 + x)^2 + 4(x^2 + x) - 12. We'll break down the steps and demonstrate the solution.

Step 1: Recognize the Pattern

Observe that the expression has a repeated term, (x^2 + x). This is a strong indicator that we can use a substitution method to simplify the factoring process.

Step 2: Substitute a Variable

Let's introduce a new variable, say 'y', to represent (x^2 + x).

The expression becomes: y^2 + 4y - 12

Step 3: Factor the Simplified Expression

The expression now resembles a standard quadratic trinomial, which can be factored as follows:

(y + 6)(y - 2)

Step 4: Substitute Back the Original Term

Now, replace 'y' with its original value, (x^2 + x):

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

Step 5: Factor the Remaining Trinomials (Optional)

The factors (x^2 + x + 6) and (x^2 + x - 2) can be further factored, although they might not result in simple linear factors.

(x^2 + x + 6) doesn't factor easily using real numbers.
(x^2 + x - 2) can be factored as (x + 2)(x - 1).

Final Factored Form:

The fully factored form of the original expression is:

(x^2 + x + 6)(x + 2)(x - 1)

Summary:

By recognizing the repeated term and using a substitution method, we effectively simplified the factoring process. We then factored the simplified expression and substituted back the original term to obtain the fully factored form.