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Factor. $\newline$ $4n^2 + 4n + 1$

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Factor quadratics: special cases

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Q. Factor.

\newline

4n^2 + 4n + 1

Check Perfect Square Trinomial: Determine if the quadratic expression can be factored as a perfect square trinomial. $\newline$ A perfect square trinomial is in the form $(a + b)^2 = a^2 + 2ab + b^2$ . We need to check if $4n^2 + 4n + 1$ fits this pattern. $\newline$ $4n^2$ can be written as $(2n)^2$ , and $1$ can be written as $(1)^2$ . The middle term, $4n$ , should be twice the product of $2n$ and $1$ if the expression is a perfect square trinomial. $\newline$ Let's check: $2 \times 2n \times 1 = 4n$ , which is indeed the middle term. $\newline$ So, $4n^2 + 4n + 1$ is a perfect square trinomial.
Factor Using Pattern: Factor the expression using the perfect square trinomial pattern. $\newline$ Since we have identified that $4n^2 + 4n + 1$ is a perfect square trinomial, it can be factored as $(2n + 1)^2$ . $\newline$ This is because $(2n + 1)(2n + 1) = 4n^2 + 2n + 2n + 1 = 4n^2 + 4n + 1$ .

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