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Factor. $\newline$ $4p^2 + 12p + 9$

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Factor quadratics: special cases

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Q. Factor.

\newline

4p^2 + 12p + 9

Check Perfect Square Trinomial: Determine if the quadratic 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 $4p^2 + 12p + 9$ fits this pattern. $\newline$ $4p^2$ can be written as $(2p)^2$ , and $9$ can be written as $3^2$ . The middle term, $12p$ , should be $2$ times the product of the square roots of the first and last terms if it is a perfect square trinomial. $\newline$ Let's check: $2 \times (2p) \times 3 = 12p$ , which matches the middle term.
Write as Perfect Square: Write the quadratic as a perfect square trinomial. $\newline$ Since the quadratic fits the pattern of a perfect square trinomial, we can write it as: $\newline$ $(2p + 3)^2 = (2p)^2 + 2\cdot(2p)\cdot3 + 3^2$ $\newline$ This matches the original expression $4p^2 + 12p + 9$ .
Factor Trinomial: Factor the perfect square trinomial. $\newline$ The factored form of the expression is simply the square of the binomial we identified: $\newline$ $(2p + 3)^2$ $\newline$ So, the factored form of $4p^2 + 12p + 9$ is $(2p + 3)(2p + 3)$ or $(2p + 3)^2$ .

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