Factor polynomial $$ 5x^{4}-7x^{3}-6x^{2} $$

$$ 5x^{4}-7x^{3}-6x^{2} = x^{2}( x-2 )( 5x+3 ) $$

Step 1 :

After factoring out $ x^{2} $ we have:

$$ 5x^{4}-7x^{3}-6x^{2} = x^{2} ( 5x^{2}-7x-6 ) $$

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:

$$ a = 5 , b = -7 ~ \text{ and } ~ c = -6 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = -6} $.

$$ a \cdot c = -30 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -30 $ and add to $ b = -7 $.

Step 5: All pairs of numbers with a product of $ -30 $ are:

PRODUCT = -30
-1     30	1     -30
-2     15	2     -15
-3     10	3     -10
-5     6	5     -6

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

PRODUCT = -30 and SUM = -7
-1     30	1     -30
-2     15	2     -15
-3     10	3     -10
-5     6	5     -6

Step 7: Replace middle term $ -7 x $ with $ 3x-10x $:

$$ 5x^{2}-7x-6 = 5x^{2}+3x-10x-6 $$

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

$$ 5x^{2}+3x-10x-6 = x\left(5x+3\right) -2\left(5x+3\right) = \left(x-2\right) \left(5x+3\right) $$

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