Question

Factor trinomial $$ \color{blue}{ 18x^2+48x-210 } $$

Answer

The factored form is $$ \color{blue}{ 18x^2+48x-210 = \left(18x-42\right)\left(x+5\right) } $$

Explanation

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:

$$ a = 18 , b = 48 ~ \text{ and } ~ c = -210 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 18 }$ by the constant term $\color{blue}{c = -210} $.

$$ a \cdot c = -3780 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -3780 $ and add to $ b = 48 $.

Step 4: All pairs of numbers with a product of $ -3780 $ are:

PRODUCT = -3780
-1 3780	1 -3780
-2 1890	2 -1890
-3 1260	3 -1260
-4 945	4 -945
-5 756	5 -756
-6 630	6 -630
-7 540	7 -540
-9 420	9 -420
-10 378	10 -378
-12 315	12 -315
-14 270	14 -270
-15 252	15 -252
-18 210	18 -210
-20 189	20 -189
-21 180	21 -180
-27 140	27 -140
-28 135	28 -135
-30 126	30 -126
-35 108	35 -108
-36 105	36 -105
-42 90	42 -90
-45 84	45 -84
-54 70	54 -70
-60 63	60 -63

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 48 }$

PRODUCT = -3780 and SUM = 48
-1 3780	1 -3780
-2 1890	2 -1890
-3 1260	3 -1260
-4 945	4 -945
-5 756	5 -756
-6 630	6 -630
-7 540	7 -540
-9 420	9 -420
-10 378	10 -378
-12 315	12 -315
-14 270	14 -270
-15 252	15 -252
-18 210	18 -210
-20 189	20 -189
-21 180	21 -180
-27 140	27 -140
-28 135	28 -135
-30 126	30 -126
-35 108	35 -108
-36 105	36 -105
-42 90	42 -90
-45 84	45 -84
-54 70	54 -70
-60 63	60 -63

Step 6: Replace middle term $ 48 x $ with $ 90x-42x $:

$$ 18x^{2}+48x-210 = 18x^{2}+90x-42x-210 $$

Step 7: Apply factoring by grouping. Factor $ 18x $ out of the first two terms and $ -42 $ out of the last two terms.

$$ 18x^{2}+90x-42x-210 = 18x\left(x+5\right) -42\left(x+5\right) = \left(18x-42\right) \left(x+5\right) $$

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