Factor polynomial $$ 3x^{4}-6x^{3}+12x^{2} $$

$$ 3x^{4}-6x^{3}+12x^{2} = 3x^{2}( x^{2}-2x+4 ) $$

Step 1 :

After factoring out $ 3x^{2} $ we have:

$$ 3x^{4}-6x^{3}+12x^{2} = 3x^{2} ( x^{2}-2x+4 ) $$

Step 2 :

Step 2: Identify constants $ \color{blue}{ b }$ and $\color{red}{ c }$. ( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:

$$ \color{blue}{ b = -2 } ~ \text{ and } ~ \color{red}{ c = 4 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -2 } $ and multiply to $ \color{red}{ 4 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 4 }$.

PRODUCT = 4
1     4	-1     -4
2     2	-2     -2

Step 4: Because none of these pairs will give us a sum of $ \color{blue}{ -2 }$, we conclude the polynomial cannot be factored.

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