(x^2+x+3)(x^2+x+4)-12

2 min read Jun 17, 2024

Factoring the Expression (x^2 + x + 3)(x^2 + x + 4) - 12

This article explores the steps involved in factoring the expression (x^2 + x + 3)(x^2 + x + 4) - 12. We'll utilize techniques like substitution and the difference of squares to simplify the expression and find its factored form.

1. Substitution

To make the factorization process easier, we can introduce a substitution. Let's represent the common expression (x^2 + x) as 'y'. This simplifies our original expression to:

(y + 3)(y + 4) - 12

2. Expanding the Expression

Expanding the expression above, we get:

y^2 + 7y + 12 - 12

Simplifying further, we have:

y^2 + 7y

3. Factoring out the Common Factor

We can factor out a 'y' from the expression:

y(y + 7)

4. Replacing the Substitution

Now, let's substitute back the original expression for 'y':

(x^2 + x)(x^2 + x + 7)

5. Final Factored Form

Therefore, the fully factored form of the expression (x^2 + x + 3)(x^2 + x + 4) - 12 is (x^2 + x)(x^2 + x + 7).

Conclusion

By using substitution and basic factoring techniques, we were able to successfully factor the given expression. This approach simplifies the process and makes it easier to identify the factors.