Question

factor the trinomial completely.\\(18n^{4}-84n^{2}+98\\)

Explanation:

Step1: Factor out GCF

The greatest common factor of $18n^4$, $-84n^2$, and $98$ is $2$.
$18n^4 - 84n^2 + 98 = 2(9n^4 - 42n^2 + 49)$

Step2: Recognize perfect square trinomial

Let $u = n^2$, so $9n^4 - 42n^2 + 49 = 9u^2 - 42u + 49$. This fits $a^2-2ab+b^2$ where $a=3u$, $b=7$.
$9u^2 - 42u + 49 = (3u - 7)^2$

Step3: Substitute back $u=n^2$

Replace $u$ with $n^2$ to get the fully factored form.
$(3u - 7)^2 = (3n^2 - 7)^2$

Step4: Combine with factored GCF

Multiply the GCF back with the squared binomial.
$2(3n^2 - 7)^2$

Answer:

$2(3n^2 - 7)^2$