Factor polynomial $$ 14t^{2}-53t+14 $$

$$ 14t^{2}-53t+14 = ( 2t-7 )( 7t-2 ) $$

Step 1: 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:

$$ a = 14 , b = -53 ~ \text{ and } ~ c = 14 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 14 }$ by the constant term $\color{blue}{c = 14} $.

$$ a \cdot c = 196 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 196 $ and add to $ b = -53 $.

Step 4: All pairs of numbers with a product of $ 196 $ are:

PRODUCT = 196
1     196	-1     -196
2     98	-2     -98
4     49	-4     -49
7     28	-7     -28
14     14	-14     -14

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -53 }$

PRODUCT = 196 and SUM = -53
1     196	-1     -196
2     98	-2     -98
4     49	-4     -49
7     28	-7     -28
14     14	-14     -14

Step 6: Replace middle term $ -53 x $ with $ -4x-49x $:

$$ 14x^{2}-53x+14 = 14x^{2}-4x-49x+14 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -7 $ out of the last two terms.

$$ 14x^{2}-4x-49x+14 = 2x\left(7x-2\right) -7\left(7x-2\right) = \left(2x-7\right) \left(7x-2\right) $$

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