Step 1 :

After factoring out $ -1 $ we have:

$$ -5x^{2}-27x-10 = - ~ ( 5x^{2}+27x+10 ) $$

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:

$$ a = 5 , b = 27 ~ \text{ and } ~ c = 10 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = 10} $.

$$ a \cdot c = 50 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 50 $ and add to $ b = 27 $.

Step 5: All pairs of numbers with a product of $ 50 $ are:

PRODUCT = 50
1     50	-1     -50
2     25	-2     -25
5     10	-5     -10

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 27 }$

PRODUCT = 50 and SUM = 27
1     50	-1     -50
2     25	-2     -25
5     10	-5     -10

Step 7: Replace middle term $ 27 x $ with $ 25x+2x $:

$$ 5x^{2}+27x+10 = 5x^{2}+25x+2x+10 $$

Step 8: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ 2 $ out of the last two terms.

$$ 5x^{2}+25x+2x+10 = 5x\left(x+5\right) + 2\left(x+5\right) = \left(5x+2\right) \left(x+5\right) $$

Polynomial Factoring Calculator