Factor polynomial $$ 25x^{2}-250x+561 $$

$$ 25x^{2}-250x+561 = ( 5x-33 )( 5x-17 ) $$

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:

$$ a = 25 , b = -250 ~ \text{ and } ~ c = 561 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 25 }$ by the constant term $\color{blue}{c = 561} $.

$$ a \cdot c = 14025 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 14025 $ and add to $ b = -250 $.

Step 4: All pairs of numbers with a product of $ 14025 $ are:

PRODUCT = 14025
1     14025	-1     -14025
3     4675	-3     -4675
5     2805	-5     -2805
11     1275	-11     -1275
15     935	-15     -935
17     825	-17     -825
25     561	-25     -561
33     425	-33     -425
51     275	-51     -275
55     255	-55     -255
75     187	-75     -187
85     165	-85     -165

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -250 }$

PRODUCT = 14025 and SUM = -250
1     14025	-1     -14025
3     4675	-3     -4675
5     2805	-5     -2805
11     1275	-11     -1275
15     935	-15     -935
17     825	-17     -825
25     561	-25     -561
33     425	-33     -425
51     275	-51     -275
55     255	-55     -255
75     187	-75     -187
85     165	-85     -165

Step 6: Replace middle term $ -250 x $ with $ -85x-165x $:

$$ 25x^{2}-250x+561 = 25x^{2}-85x-165x+561 $$

Step 7: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ -33 $ out of the last two terms.

$$ 25x^{2}-85x-165x+561 = 5x\left(5x-17\right) -33\left(5x-17\right) = \left(5x-33\right) \left(5x-17\right) $$

Polynomial Factoring Calculator