Factorise completely:

1)3g-3t+2mg-2mt
2)3x squared + 2x-8
3)3x squared-27

(G-T) ( 3 + 2m)

(3x+4) (x-2)

3 (x+3) ( x-3)

1. a "grouping" question

3g-3t+2mg-2mt
= 3(g-t) + 2m(g-t)
You do the last step

2. 3x^2 + 2x-8
= (3x - 4)(? + ?) can you figure out the second bracket?

3. 3x^2 - 27
= 3(x^2 - 9)
= ..... mmhh, looks like a difference of squares

To factorize completely, we need to find the common factors between the terms and then use the factoring techniques for each equation:

1) 3g - 3t + 2mg - 2mt
= 3(g - t) + 2m(g - t)
= (g - t)(3 + 2m)

2) 3x^2 + 2x - 8
This quadratic equation cannot be factored further using integer factors. You can, however, use the quadratic formula to find the roots of the equation.

3) 3x^2 - 27
= 3(x^2 - 9)
= 3(x + 3)(x - 3)

So, the completely factorized forms are:
1) (g - t)(3 + 2m)
2) 3x^2 + 2x - 8 (No further factorization)
3) 3(x + 3)(x - 3)

To factorize completely, we need to find common factors and then factor out those common factors from each term.

1) Factorize completely: 3g - 3t + 2mg - 2mt

Step 1: Group the terms with common factors.
In this expression, we have two common factors: 3 and (t - g).
Grouping the terms, we have:
3(g - t) + 2m(g - t)

Step 2: Factor out the common factor.
The expression now becomes:
(3 + 2m)(g - t)

2) Factorize completely: 3x^2 + 2x - 8

Step 1: Look for factors of the quadratic expression.
In this case, we notice that the expression can be factored using the grouping or the quadratic formula method.

Step 2: Factor using grouping or quadratic formula.
We can use the quadratic formula to find the roots of the quadratic equation. The quadratic formula is given as:
x = (-b ± √(b^2 - 4ac)) / (2a)

For the given expression, a = 3, b = 2, and c = -8.
Using the quadratic formula, we find the roots as:
x = ( -2 ± √(2^2 - 4 * 3 * -8)) / (2 * 3)
x = ( -2 ± √(4 + 96)) / 6
x = ( -2 ± √100) / 6
x = ( -2 ± 10) / 6

So the roots of the quadratic equation are x = 2/3 and x = -4/3.

Therefore, the quadratic expression can be factored as:
3x^2 + 2x - 8 = 3(x - 2/3)(x + 4/3)

3) Factorize completely: 3x^2 - 27

Step 1: Look for the greatest common factor (GCF).
In this expression, the GCF is 3.

Step 2: Factor out the GCF.
We can factor out 3 from both terms, and the expression becomes:
3(x^2 - 9)

Step 3: Factor the remaining expression.
The expression x^2 - 9 is a difference of squares, which can be factored as:
x^2 - 9 = (x + 3)(x - 3)

Therefore, the complete factorization of 3x^2 - 27 is:
3(x + 3)(x - 3)