Question

factor completely.
$16n^2 + 16n + 4$

Explanation:

Step1: Identify common factor

The given expression is \(16n^2 + 16n + 4\). We can see that each term is divisible by 4. So, factor out 4 first.
\(16n^2 + 16n + 4 = 4(4n^2 + 4n + 1)\)

Step2: Factor the quadratic

Now, look at the quadratic \(4n^2 + 4n + 1\). It is a perfect square trinomial. The form of a perfect square trinomial is \(a^2 + 2ab + b^2=(a + b)^2\). Here, \(a = 2n\) (since \((2n)^2 = 4n^2\)) and \(b = 1\) (since \(1^2 = 1\)), and \(2ab = 2\times2n\times1 = 4n\), which matches the middle term. So, \(4n^2 + 4n + 1=(2n + 1)^2\)

Step3: Combine the factors

Putting it all together, the completely factored form is \(4(2n + 1)^2\)

Answer:

\(4(2n + 1)^2\)