(x2−4)(x2+6x+9)

2 min read Jun 17, 2024

Factoring and Simplifying the Expression (x² - 4)(x² + 6x + 9)

This expression represents the product of two quadratic expressions. To simplify it, we can factor each quadratic expression and then multiply the resulting factors.

Factoring the First Expression: (x² - 4)

This expression is a difference of squares, which factors as follows:

(a² - b²) = (a + b)(a - b)

Applying this to (x² - 4), we get:

(x² - 4) = (x + 2)(x - 2)

Factoring the Second Expression: (x² + 6x + 9)

This expression is a perfect square trinomial, which factors as follows:

(a² + 2ab + b²) = (a + b)²

Applying this to (x² + 6x + 9), we get:

(x² + 6x + 9) = (x + 3)²

Multiplying the Factored Expressions

Now we have:

(x² - 4)(x² + 6x + 9) = (x + 2)(x - 2)(x + 3)²

This is the fully factored form of the original expression.

Expanding the Expression (Optional)

While the factored form is often the most useful, we can also expand the expression to obtain a polynomial in standard form:

(x + 2)(x - 2)(x + 3)² = (x² - 4)(x² + 6x + 9)

Expanding the product, we get:

(x² - 4)(x² + 6x + 9) = x⁴ + 6x³ + 5x² - 24x - 36

This is the simplified polynomial form of the expression.

Conclusion

The expression (x² - 4)(x² + 6x + 9) can be simplified by factoring each quadratic expression. This gives us the factored form: (x + 2)(x - 2)(x + 3)², which can be further expanded to obtain the polynomial form: x⁴ + 6x³ + 5x² - 24x - 36.