Question

factor ( x^4 + 6x^2 + 9 ) completely. all factors in your answer should have integer coefficients.

Explanation:

Step1: Recognize the quadratic form

Let \( y = x^2 \), then the expression \( x^4 + 6x^2 + 9 \) becomes \( y^2 + 6y + 9 \).

Step2: Factor the quadratic in y

The quadratic \( y^2 + 6y + 9 \) is a perfect square trinomial, which factors as \( (y + 3)^2 \).

Step3: Substitute back \( y = x^2 \)

Substituting \( y = x^2 \) back into \( (y + 3)^2 \), we get \( (x^2 + 3)^2 \).

Answer:

\( (x^2 + 3)^2 \)