Factor polynomial $$ 8x^{2}+33x+4 $$

$$ 8x^{2}+33x+4 = ( 8x+1 )( x+4 ) $$

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:

$$ a = 8 , b = 33 ~ \text{ and } ~ c = 4 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 8 }$ by the constant term $\color{blue}{c = 4} $.

$$ a \cdot c = 32 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 32 $ and add to $ b = 33 $.

Step 4: All pairs of numbers with a product of $ 32 $ are:

PRODUCT = 32
1     32	-1     -32
2     16	-2     -16
4     8	-4     -8

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 33 }$

PRODUCT = 32 and SUM = 33
1     32	-1     -32
2     16	-2     -16
4     8	-4     -8

Step 6: Replace middle term $ 33 x $ with $ 32x+x $:

$$ 8x^{2}+33x+4 = 8x^{2}+32x+x+4 $$

Step 7: Apply factoring by grouping. Factor $ 8x $ out of the first two terms and $ 1 $ out of the last two terms.

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

Polynomial Factoring Calculator