Question

factor trinomial completely of the type ax^2+bx+c.

-9+18x^2+21x

I am not able to solve the solution since the steps are quite confusing any help is great

Answer 1

first arrange them in the standard order
18x^2 + 21x - 9

look for a common factor, in this case 3

18x^2 + 21x - 9
= 3(6x^2 + 7x - 3)

A method which seems to be popular these days is "decomposition", (not my personal favourite)

Step1: multiply the (a)(c)
(6)(-3) = -18
step2: look for 2 factors which multiply to get -15, and whose sum is the middle term coefficent
-18 = 6(-3), 6-3 = 3 , no
= 2(-9) , 2 - 9 = -7 , no
= -2(9) , -2+9 = 7 , yeahhh the sum is 7

If you cannot find such a combination, the quadratic does not factor

step3: split the middle term into the sum of these two
3(6x^2 - 2x + 9x - 3)
= 3( 2x(3x - 1) + 3(3x - 1) )
notice that 3x-1 is a common factor, this MUST happen, if not, you made a mistake
= 3(3x-1)(2x+3)

Answer 2

small typo

should have said:

step2: look for 2 factors which multiply to get -18, and whose sum is the middle term coefficient

But you probably figured that out anyway.

Answer 3

Thanks for your help!

I understand it much better.