Step 1 :

After factoring out $ -1 $ we have:

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

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:

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

Step 3: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = -1} $.

$$ a \cdot c = -5 $$

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

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

PRODUCT = -5
-1     5	1     -5

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -4 }$

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

Step 7: Replace middle term $ -4 x $ with $ x-5x $:

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

Step 8: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -1 $ out of the last two terms.

$$ 5x^{2}+x-5x-1 = x\left(5x+1\right) -1\left(5x+1\right) = \left(x-1\right) \left(5x+1\right) $$

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