The factored form of the trinomial is:
144x^2 + 216x + 81 = (4x + 6)^2 - 135
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Explanation:
To factor the trinomial 144x^2 + 216x + 81, we can use the factoring technique called "completing the square". The steps are as follows:
Divide all the terms by the common factor 9, which gives:
16x^2 + 24x + 9
Take half of the coefficient of the x-term, square it, and add and subtract it to the expression. In this case, half of 24 is 12, so we add and subtract 12^2 = 144:
16x^2 + 24x + 144 - 144 + 9
Group the first three terms and factor it as a perfect square trinomial, and simplify the last two terms:
(4x + 6)^2 - 135
Therefore, the factored form of the trinomial is:
144x^2 + 216x + 81 = (4x + 6)^2 - 135
We can check that this is the correct factorization by expanding the squared term:
(4x + 6)^2 - 135 = (4x + 6)(4x + 6) - 135
= 16x^2 + 48x + 36 - 135
= 16x^2 + 24x + 9
Therefore, the factorization is correct.