How do I factor the following equations by grouping?

Question

4r^2-12rs-6s+4r+9s^2

a^2-4ab-4b^2-a+2b

t^2-s^2+rs-rt

Answer 1

(1) 4r^2 - 12rs - 6s + 4r + 9s^2 may be grouped as:

4r^2 - 12rs - 6s + 4r + 9s^2

= (4r^2 - 12rs + 9s^2) - 6s + 4r

= (2r - 3s)^2 + 2(2r - 3s)

= (2r - 3s)(2r - 3s + 2) are the factors.

(2) In a^2 - 4ab - 4b^2 - a + 2b, - 4b^2 should have +ve sign and may then be written as, after grouping:

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

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

= (a - 2b)(a - 2b - 1) are the desired factors.

(3) t^2 - s^2 + rs - rt may be grouped & written as:

(t^2 - s^2) + (rs - rt)

= (t - s)(t + s) - r(t - s)

= (t - s)(t + s -r) are the factors.

Answer 2

t^2 - s^2 + rs - rt

(t +s)(t-s) + r(s-t)

(t - s) [ (t+s) -r]

Answer 3

(r-3s+1)(2+3s)(4r-3s)

(a-4b-1)(2b+1)(a-2b)

(t-r)(r-s)(t-s)