Factorize: 3x^2- 7x + 2?

(x-2) (3 x-1)

3x^2- 7x + 2
3 * 2 = 6 (the product of the coefficient of x^2 and the constant)
Need to find 2 numbers that add together to make -7 (coefficient of x) and times together to make 6 (found above).
So this would be -6 and -1.
Rewrite equation as 3x^2 - 6x - x + 2 (splitting the -7x into the numbers found above)
Then factorize each half: 3x(x - 2) - 1(x - 2)
The bracket numbers form half, and the numbers in front of the brackets form the other bracket giving:
(3x - 1)(x - 2)

This can be used for any quadratic with a number in front of x^2.

For a quadratic equation such as the one above, you will have two factors in the form of (ax + b)... note, this is just the form for the factors. When we actually apply this form, we get:
(Mx + P)(Nx + Q)

When we multiply these two factors, the result is:
(MNx^2) + (MQx) +
(PNx) + (PQ)
I wrote this out on two lines to better demonstrate how I multiplied both terms in the second factor first by the first term of the first factor, then by the second term of the first factor. You can actually multiply the terms in any order, but you must multiply both terms of each factor by both terms of the other.
Placing these in a single line:
(MNx^2) + (MQx) + (PNx) + (PQ)
Next we can simplify this expression by adding the coefficients of the same variable with the same exponent:
(MNx^2) + (MQ + PN)x + (PQ)

From the original expression:
MN = 3
PQ = 2
MQ + PN = -7

If we assume that the component factors of the coefficients are all whole numbers (we can not always assume this, but it is often the case for homework assignments), we get the following possibilities:
MN = (1)(3) or (-1)(-3)
PQ = (1)(2) or (-1)(-2)
note that this means there can be FOUR possibilities for each pair, since the combinations can occur in any order.

Solving for MQ + PN = -7, the easy possibilities for the two terms would either be -1 + -6 or -3 + -4 (again, these can be in any order). The second pair can be ruled out since neither MQ nor PN have any combinations that can produce 4. However, 1 + 6 is quite easy.
MQ + PN = -7
MQ = (-3)(2) = 6
NP = (-1)(1) = 1

MQ = (3)(-2) = 6
NP = (1)(-1) = 1
Note that this time, order IS important!

Thus:
(Mx + P)(Nx + Q)=
(-3x + 1)(-x + 2) = (1 - 3x)(2 - x)
or
(3x -1)(x -2)

IMPORTANT: note that there are TWO possible answers!

You can multiply these out to verify the answer.

3x^2 - 7x + 2
= 3x^2 - x - 6x + 2
= (3x^2 - x) - (6x - 2)
= x(3x - 1) - 2(3x - 1)
= (3x - 1)(x - 2)

what two numbers multiply to get +6 and add together to give -7?
(-6 and -1)
=3x^2-6x-x+2
=3x(x-2) -1(x-2)
=(3x-1) (x-2)

you can always check your answer by multiplying this out/ foiling again

Splitting 7x into 2 gives
3x^2-6x-x+2
3x(x-2)-1(x-2)
(x-2)(3x-1)

3 * 2 = 6
and
-6 * (-1) = 6 and -6 - 1 = -7
so...
3x^2 - 6x - 1x + 2
3x(x -2) -1(x -2)
Cancel one "(x -2)"
Then
(3x -1) (x -2) ta-da

( 3 x - 1 ) ( x - 2 )

x=6

x(3x - 7) + 2