Step 1 :

After factoring out $ 5x^{3} $ we have:

$$ 10x^{5}+35x^{4}+25x^{3} = 5x^{3} ( 2x^{2}+7x+5 ) $$

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:

$$ a = 2 , b = 7 ~ \text{ and } ~ c = 5 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = 5} $.

$$ a \cdot c = 10 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 10 $ and add to $ b = 7 $.

Step 5: All pairs of numbers with a product of $ 10 $ are:

PRODUCT = 10
1     10	-1     -10
2     5	-2     -5

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 7 }$

PRODUCT = 10 and SUM = 7
1     10	-1     -10
2     5	-2     -5

Step 7: Replace middle term $ 7 x $ with $ 5x+2x $:

$$ 2x^{2}+7x+5 = 2x^{2}+5x+2x+5 $$

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

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

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