Factorise (a - b)^2 - 4(a - b)?

Both terms have a (a - b) in common, so we can pull that out:

(a - b)((a - b) - 4) =

Answer: (a - b)(a - b - 4).

-John

Factor:
(a - b)² - 4(a - b) = 0
(a - b)(a - b) = 4(a - b)
a - b = (4[a - b])/(a - b)
a - b = 4

Answer: a - b = 4

From this you can proceed to a = b + 4 and b = a - 4.

(a-b)(a-b)-4(a-b)=(a-b)(a-b-4)

To factorise, you need to divide the whole thing by the largest common factor.

The largest common factor is, in this case, (a - b)

Another way of writing your expression is:
(a - b)(a - b) - 4(a - b)
If you divide the whole thing by (a - b), you get
(a - b) - 4

so the factorised expression would be (a - b) multiplied by the above:

(a - b)[(a - b) - 4]

(a-b)^2 - 4(a-b)

= (a-b)(a-b) - 4(a-b)

The first and second terms have the common factor (a-b) so we can write:

(a-b)[(a-b) - 4]

This is the original expression now factorised.

(a - b)(a - b) - 4(a - b)
(a - b) [ (a - b) - 4 ]
(a - b) [ a - b - 4 ]

(a-b)² - 4(a-b), the whole thing is divisible by (a-b)?

(a-b)² / (a-b) = (a-b)
-4(a-b) / (a-b) = -4


(a-b)(a-b-4)

(a - b)^2 - 4(a - b)
= (a - b)(a - b) - 4(a - b)
= (a - b)(a - b - 4)

(a - b)^2 - 4(a - b)
a^2-2ab+b^2-4a+4b
(a^2-4a+4)+(b^2+4b+4)-2ab-8
(a-2)^2+(b+2)^2-2(ab+4)

Notice that with (a-b)² - 4(a-b), the whole thing is divisible by (a-b)?

(a-b)² / (a-b) = (a-b)
-4(a-b) / (a-b) = -4

Because you have established this, you can now "bring out" the (a-b):

(a-b)(a-b-4)