Factor polynomial $$ 3x^{2}-118x-765 $$
$$ 3x^{2}-118x-765 = ( x-45 )( 3x+17 ) $$
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:
Step 2: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = -765} $.
$$ a \cdot c = -2295 $$Step 3: Find out two numbers that multiply to $ a \cdot c = -2295 $ and add to $ b = -118 $.
Step 4: All pairs of numbers with a product of $ -2295 $ are:
| PRODUCT = -2295 | |
| -1 2295 | 1 -2295 |
| -3 765 | 3 -765 |
| -5 459 | 5 -459 |
| -9 255 | 9 -255 |
| -15 153 | 15 -153 |
| -17 135 | 17 -135 |
| -27 85 | 27 -85 |
| -45 51 | 45 -51 |
Step 5: Find out which factor pair sums up to $\color{blue}{ b = -118 }$
| PRODUCT = -2295 and SUM = -118 | |
| -1 2295 | 1 -2295 |
| -3 765 | 3 -765 |
| -5 459 | 5 -459 |
| -9 255 | 9 -255 |
| -15 153 | 15 -153 |
| -17 135 | 17 -135 |
| -27 85 | 27 -85 |
| -45 51 | 45 -51 |
Step 6: Replace middle term $ -118 x $ with $ 17x-135x $:
$$ 3x^{2}-118x-765 = 3x^{2}+17x-135x-765 $$Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -45 $ out of the last two terms.
$$ 3x^{2}+17x-135x-765 = x\left(3x+17\right) -45\left(3x+17\right) = \left(x-45\right) \left(3x+17\right) $$