Factor polynomial $$ 77x^{2}-155x+72 $$
$$ 77x^{2}-155x+72 = ( 7x-9 )( 11x-8 ) $$
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 $\color{blue}{ a = 77 }$ by the constant term $\color{blue}{c = 72} $.
$$ a \cdot c = 5544 $$Step 3: Find out two numbers that multiply to $ a \cdot c = 5544 $ and add to $ b = -155 $.
Step 4: All pairs of numbers with a product of $ 5544 $ are:
| PRODUCT = 5544 | |
| 1 5544 | -1 -5544 |
| 2 2772 | -2 -2772 |
| 3 1848 | -3 -1848 |
| 4 1386 | -4 -1386 |
| 6 924 | -6 -924 |
| 7 792 | -7 -792 |
| 8 693 | -8 -693 |
| 9 616 | -9 -616 |
| 11 504 | -11 -504 |
| 12 462 | -12 -462 |
| 14 396 | -14 -396 |
| 18 308 | -18 -308 |
| 21 264 | -21 -264 |
| 22 252 | -22 -252 |
| 24 231 | -24 -231 |
| 28 198 | -28 -198 |
| 33 168 | -33 -168 |
| 36 154 | -36 -154 |
| 42 132 | -42 -132 |
| 44 126 | -44 -126 |
| 56 99 | -56 -99 |
| 63 88 | -63 -88 |
| 66 84 | -66 -84 |
| 72 77 | -72 -77 |
Step 5: Find out which factor pair sums up to $\color{blue}{ b = -155 }$
| PRODUCT = 5544 and SUM = -155 | |
| 1 5544 | -1 -5544 |
| 2 2772 | -2 -2772 |
| 3 1848 | -3 -1848 |
| 4 1386 | -4 -1386 |
| 6 924 | -6 -924 |
| 7 792 | -7 -792 |
| 8 693 | -8 -693 |
| 9 616 | -9 -616 |
| 11 504 | -11 -504 |
| 12 462 | -12 -462 |
| 14 396 | -14 -396 |
| 18 308 | -18 -308 |
| 21 264 | -21 -264 |
| 22 252 | -22 -252 |
| 24 231 | -24 -231 |
| 28 198 | -28 -198 |
| 33 168 | -33 -168 |
| 36 154 | -36 -154 |
| 42 132 | -42 -132 |
| 44 126 | -44 -126 |
| 56 99 | -56 -99 |
| 63 88 | -63 -88 |
| 66 84 | -66 -84 |
| 72 77 | -72 -77 |
Step 6: Replace middle term $ -155 x $ with $ -56x-99x $:
$$ 77x^{2}-155x+72 = 77x^{2}-56x-99x+72 $$Step 7: Apply factoring by grouping. Factor $ 7x $ out of the first two terms and $ -9 $ out of the last two terms.
$$ 77x^{2}-56x-99x+72 = 7x\left(11x-8\right) -9\left(11x-8\right) = \left(7x-9\right) \left(11x-8\right) $$