Factor polynomial $$ -16x^{2}+32x+9 $$

$$ -16x^{2}+32x+9 = -( 4x-9 )( 4x+1 ) $$

Step 1 :

After factoring out $ -1 $ we have:

$$ -16x^{2}+32x+9 = - ~ ( 16x^{2}-32x-9 ) $$

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:

$$ a = 16 , b = -32 ~ \text{ and } ~ c = -9 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 16 }$ by the constant term $\color{blue}{c = -9} $.

$$ a \cdot c = -144 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -144 $ and add to $ b = -32 $.

Step 5: All pairs of numbers with a product of $ -144 $ are:

PRODUCT = -144
-1     144	1     -144
-2     72	2     -72
-3     48	3     -48
-4     36	4     -36
-6     24	6     -24
-8     18	8     -18
-9     16	9     -16
-12     12	12     -12

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -32 }$

PRODUCT = -144 and SUM = -32
-1     144	1     -144
-2     72	2     -72
-3     48	3     -48
-4     36	4     -36
-6     24	6     -24
-8     18	8     -18
-9     16	9     -16
-12     12	12     -12

Step 7: Replace middle term $ -32 x $ with $ 4x-36x $:

$$ 16x^{2}-32x-9 = 16x^{2}+4x-36x-9 $$

Step 8: Apply factoring by grouping. Factor $ 4x $ out of the first two terms and $ -9 $ out of the last two terms.

$$ 16x^{2}+4x-36x-9 = 4x\left(4x+1\right) -9\left(4x+1\right) = \left(4x-9\right) \left(4x+1\right) $$

Polynomial Factoring Calculator