Factor polynomial $$ 4y^{2}-5y+1 $$

$$ 4y^{2}-5y+1 = ( y-1 )( 4y-1 ) $$

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:

$$ a = 4 , b = -5 ~ \text{ and } ~ c = 1 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = 1} $.

$$ a \cdot c = 4 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 4 $ and add to $ b = -5 $.

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

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

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

PRODUCT = 4 and SUM = -5
1     4	-1     -4
2     2	-2     -2

Step 6: Replace middle term $ -5 x $ with $ -x-4x $:

$$ 4x^{2}-5x+1 = 4x^{2}-x-4x+1 $$

Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -1 $ out of the last two terms.

$$ 4x^{2}-x-4x+1 = x\left(4x-1\right) -1\left(4x-1\right) = \left(x-1\right) \left(4x-1\right) $$

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