Factor polynomial $$ 36t^{4}-144t^{3}+156t^{2} $$

$$ 36t^{4}-144t^{3}+156t^{2} = 12t^{2}( 3t^{2}-12t+13 ) $$

Step 1 :

After factoring out $ 12t^{2} $ we have:

$$ 36t^{4}-144t^{3}+156t^{2} = 12t^{2} ( 3t^{2}-12t+13 ) $$

Step 2 :

Step 2: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 3 , b = -12 ~ \text{ and } ~ c = 13 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = 13} $.

$$ a \cdot c = 39 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 39 $ and add to $ b = -12 $.

Step 5: All pairs of numbers with a product of $ 39 $ are:

PRODUCT = 39
1     39	-1     -39
3     13	-3     -13

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -12 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ -12 }$, we conclude the polynomial cannot be factored.

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