Factor polynomial $$ 11x^{2}+15x-2700 $$
$$ 11x^{2}+15x-2700 = ( x-15 )( 11x+180 ) $$
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 $\color{blue}{ a = 11 }$ by the constant term $\color{blue}{c = -2700} $.
$$ a \cdot c = -29700 $$Step 3: Find out two numbers that multiply to $ a \cdot c = -29700 $ and add to $ b = 15 $.
Step 4: All pairs of numbers with a product of $ -29700 $ are:
| PRODUCT = -29700 | |
| -1 29700 | 1 -29700 |
| -2 14850 | 2 -14850 |
| -3 9900 | 3 -9900 |
| -4 7425 | 4 -7425 |
| -5 5940 | 5 -5940 |
| -6 4950 | 6 -4950 |
| -9 3300 | 9 -3300 |
| -10 2970 | 10 -2970 |
| -11 2700 | 11 -2700 |
| -12 2475 | 12 -2475 |
| -15 1980 | 15 -1980 |
| -18 1650 | 18 -1650 |
| -20 1485 | 20 -1485 |
| -22 1350 | 22 -1350 |
| -25 1188 | 25 -1188 |
| -27 1100 | 27 -1100 |
| -30 990 | 30 -990 |
| -33 900 | 33 -900 |
| -36 825 | 36 -825 |
| -44 675 | 44 -675 |
| -45 660 | 45 -660 |
| -50 594 | 50 -594 |
| -54 550 | 54 -550 |
| -55 540 | 55 -540 |
| -60 495 | 60 -495 |
| -66 450 | 66 -450 |
| -75 396 | 75 -396 |
| -90 330 | 90 -330 |
| -99 300 | 99 -300 |
| -100 297 | 100 -297 |
| -108 275 | 108 -275 |
| -110 270 | 110 -270 |
| -132 225 | 132 -225 |
| -135 220 | 135 -220 |
| -150 198 | 150 -198 |
| -165 180 | 165 -180 |
Step 5: Find out which factor pair sums up to $\color{blue}{ b = 15 }$
| PRODUCT = -29700 and SUM = 15 | |
| -1 29700 | 1 -29700 |
| -2 14850 | 2 -14850 |
| -3 9900 | 3 -9900 |
| -4 7425 | 4 -7425 |
| -5 5940 | 5 -5940 |
| -6 4950 | 6 -4950 |
| -9 3300 | 9 -3300 |
| -10 2970 | 10 -2970 |
| -11 2700 | 11 -2700 |
| -12 2475 | 12 -2475 |
| -15 1980 | 15 -1980 |
| -18 1650 | 18 -1650 |
| -20 1485 | 20 -1485 |
| -22 1350 | 22 -1350 |
| -25 1188 | 25 -1188 |
| -27 1100 | 27 -1100 |
| -30 990 | 30 -990 |
| -33 900 | 33 -900 |
| -36 825 | 36 -825 |
| -44 675 | 44 -675 |
| -45 660 | 45 -660 |
| -50 594 | 50 -594 |
| -54 550 | 54 -550 |
| -55 540 | 55 -540 |
| -60 495 | 60 -495 |
| -66 450 | 66 -450 |
| -75 396 | 75 -396 |
| -90 330 | 90 -330 |
| -99 300 | 99 -300 |
| -100 297 | 100 -297 |
| -108 275 | 108 -275 |
| -110 270 | 110 -270 |
| -132 225 | 132 -225 |
| -135 220 | 135 -220 |
| -150 198 | 150 -198 |
| -165 180 | 165 -180 |
Step 6: Replace middle term $ 15 x $ with $ 180x-165x $:
$$ 11x^{2}+15x-2700 = 11x^{2}+180x-165x-2700 $$Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -15 $ out of the last two terms.
$$ 11x^{2}+180x-165x-2700 = x\left(11x+180\right) -15\left(11x+180\right) = \left(x-15\right) \left(11x+180\right) $$