factor each trinomial completely and explain the steps to get there
-3a^2b + 12ab - 12b
3x^4 + 24x^3 + 48x^2
For the trinomial -3a^2b + 12ab - 12b, we want to factor out the greatest common factor from all three terms. In this case, the greatest common factor is -3b. So, we can rewrite the trinomial as:
-3b(a^2 - 4a + 4)
Next, we want to factor the quadratic expression a^2 - 4a + 4. This quadratic expression can be factored into:
(a - 2)(a - 2)
Therefore, the completely factored form of the trinomial -3a^2b + 12ab - 12b is:
-3b(a - 2)(a - 2)
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For the trinomial 3x^4 + 24x^3 + 48x^2, we want to factor out the greatest common factor from all three terms. In this case, the greatest common factor is 3x^2. So, we can rewrite the trinomial as:
3x^2(x^2 + 8x + 16)
Next, we want to factor the quadratic expression x^2 + 8x + 16. This quadratic expression can be factored into:
(x + 4)(x + 4) or (x + 4)^2
Therefore, the completely factored form of the trinomial 3x^4 + 24x^3 + 48x^2 is:
3x^2(x + 4)^2