Question

factor $x^{4}+12x^{2}+36$ completely.
all factors in your answer should have integer coefficients.

Explanation:

Step1: Recognize perfect square trinomial

Let $y = x^2$, rewrite the expression:
$y^2 + 12y + 36$

Step2: Factor the quadratic trinomial

Use the perfect square formula $a^2+2ab+b^2=(a+b)^2$, where $a=y$, $b=6$:
$y^2 + 12y + 36=(y+6)^2$

Step3: Substitute back $y=x^2$

Replace $y$ with $x^2$:
$(x^2+6)^2$

Answer:

$(x^2 + 6)^2$