Factor polynomial $$ -5x^{2}+23x+10 $$

$$ -5x^{2}+23x+10 = -( x-5 )( 5x+2 ) $$

Step 1 :

After factoring out $ -1 $ we have:

$$ -5x^{2}+23x+10 = - ~ ( 5x^{2}-23x-10 ) $$

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:

$$ a = 5 , b = -23 ~ \text{ and } ~ c = -10 $$

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 = -23 $.

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 $\color{blue}{ b = -23 }$

PRODUCT = -50 and SUM = -23
-1     50	1     -50
-2     25	2     -25
-5     10	5     -10

Step 7: Replace middle term $ -23 x $ with $ 2x-25x $:

$$ 5x^{2}-23x-10 = 5x^{2}+2x-25x-10 $$

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

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

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