Factor polynomial $$ 24x^{2}-29x-4 $$

$$ 24x^{2}-29x-4 = ( 3x-4 )( 8x+1 ) $$

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:

$$ a = 24 , b = -29 ~ \text{ and } ~ c = -4 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 24 }$ by the constant term $\color{blue}{c = -4} $.

$$ a \cdot c = -96 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -96 $ and add to $ b = -29 $.

Step 4: All pairs of numbers with a product of $ -96 $ are:

PRODUCT = -96
-1     96	1     -96
-2     48	2     -48
-3     32	3     -32
-4     24	4     -24
-6     16	6     -16
-8     12	8     -12

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -29 }$

PRODUCT = -96 and SUM = -29
-1     96	1     -96
-2     48	2     -48
-3     32	3     -32
-4     24	4     -24
-6     16	6     -16
-8     12	8     -12

Step 6: Replace middle term $ -29 x $ with $ 3x-32x $:

$$ 24x^{2}-29x-4 = 24x^{2}+3x-32x-4 $$

Step 7: Apply factoring by grouping. Factor $ 3x $ out of the first two terms and $ -4 $ out of the last two terms.

$$ 24x^{2}+3x-32x-4 = 3x\left(8x+1\right) -4\left(8x+1\right) = \left(3x-4\right) \left(8x+1\right) $$

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