Factor polynomial $$ 5x^{2}-2x-3 $$

$$ 5x^{2}-2x-3 = ( x-1 )( 5x+3 ) $$

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:

$$ a = 5 , b = -2 ~ \text{ and } ~ c = -3 $$

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

$$ a \cdot c = -15 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -15 $ and add to $ b = -2 $.

Step 4: All pairs of numbers with a product of $ -15 $ are:

PRODUCT = -15
-1     15	1     -15
-3     5	3     -5

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -2 }$

PRODUCT = -15 and SUM = -2
-1     15	1     -15
-3     5	3     -5

Step 6: Replace middle term $ -2 x $ with $ 3x-5x $:

$$ 5x^{2}-2x-3 = 5x^{2}+3x-5x-3 $$

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

$$ 5x^{2}+3x-5x-3 = x\left(5x+3\right) -1\left(5x+3\right) = \left(x-1\right) \left(5x+3\right) $$

Polynomial Factoring Calculator