How to factorise by grouping?

1. first thy to factorise based on x and y.

ax + 2ay + 5x + 10y ===> x (a + 5) + y (2a + 10)

then you will notice that there is also another 2 which can be factorised in the y group.

x (a + 5) + y (2a +10) ====> x (a + 5) + 2y (a + 5)

now if you have noticed you can factorise (a + 5)

x (a + 5) + 2y (a + 5) ===> (a + 5) (x + 2y)

2. in this case first factorise x from x^2 and -3x

x^2 - 3x + 4x - 12 ===> x (x - 3) + 4x - 12

then factorise 4 from +4x and -12

x (x - 3) + 4x - 12 ===> x (x - 3) + 4 (x - 3)

now you can see that you can factorise (x - 3)

x (x - 3) + 4 (x - 3) ===> (x - 3) (x + 4)

Hint : to solve these sort of problems, the best is to try to find common relation between them, then group them based on their common relation and factorise on that bases.

I hope it helped.:)

You can do the first one two ways, but the answer is the same:

ax + 2ay + 5x +10y = a(x + 2y) + 5(x+2y) = (a + 5)(x + 2y)
ax + 2ay + 5x +10y = x(a + 5) + 2y(a + 5) = (a + 5)(x + 2y)

You do the second one by simple factoring

x^2 - 3x + 4x - 12 = x^2 + x - 12 = (x + 4)(x - 3)

1)
ax + 2ay + 5x + 10y
= (ax + 2ay) + (5x + 10y)
= a(x + 2y) + 5(x + 2y)
= (x + 2y)(a + 5)

2)
x^2 - 3x + 4x - 12
= (x^2 - 3x) + (4x - 12)
= x(x - 3) + 4(x - 3)
= (x - 3)(x + 4)

look at each pair and take out a common factor

(ax + 2ay) + (5x + 10y)
a(x + 2y) + 5(x + 2y)

now take out the common (x + 2y)
(x + 2y)(a + 5)

(x^2 - 3x) + (4x - 12)
x(x - 3) + 4(x - 3)
(x - 3)(x + 4)