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Answer :
Final answer:
The polynomial 12x^4 + 18x^3 + 45x^2 can be factored completely by identifying the common factor among all the terms and factoring it out from the original polynomial. The final factored form is 3x^2(4x^2 + 6x + 15).
Explanation:
The question asks us to factor the polynomial 12x4 + 18x3 + 45x2 completely. The first step in factoring this expression is to identify any common factors among all of the terms. In this case, each term shares a common factor of 3x2. We can factor that term out of the original polynomial to get 3x^2(4x^2 + 6x + 15).
Then, we can observe if the expression inside the brackets can be further factored. However, the quadratic expression 4x^2 + 6x + 15 cannot be factored further into a product of binomials (it doesn't have real roots), so our final answer is 3x^2(4x^2 + 6x + 15).
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