Factor polynomial $$ -7a^{2}+186a+81 $$

$$ -7a^{2}+186a+81 = -( a-27 )( 7a+3 ) $$

Step 1 :

After factoring out $ -1 $ we have:

$$ -7a^{2}+186a+81 = - ~ ( 7a^{2}-186a-81 ) $$

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:

$$ a = 7 , b = -186 ~ \text{ and } ~ c = -81 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 7 }$ by the constant term $\color{blue}{c = -81} $.

$$ a \cdot c = -567 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -567 $ and add to $ b = -186 $.

Step 5: All pairs of numbers with a product of $ -567 $ are:

PRODUCT = -567
-1     567	1     -567
-3     189	3     -189
-7     81	7     -81
-9     63	9     -63
-21     27	21     -27

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -186 }$

PRODUCT = -567 and SUM = -186
-1     567	1     -567
-3     189	3     -189
-7     81	7     -81
-9     63	9     -63
-21     27	21     -27

Step 7: Replace middle term $ -186 x $ with $ 3x-189x $:

$$ 7x^{2}-186x-81 = 7x^{2}+3x-189x-81 $$

Step 8: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -27 $ out of the last two terms.

$$ 7x^{2}+3x-189x-81 = x\left(7x+3\right) -27\left(7x+3\right) = \left(x-27\right) \left(7x+3\right) $$

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