Factoring is the process of finding the factors of a given expression. It is a very useful tool in mathematics that helps simplify expressions and solve equations. In the context provided, there are four expressions that need to be factored.

Expression 1: X-4 = (x+2) (x-2)

This expression can be factored using the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b). In this case, a = x and b = 2. Therefore, we can write:

x^2 - 2^2 = (x + 2)(x - 2)

Simplifying the left-hand side gives:

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

Expression 2: 2 (11√ x²=2x-3 = (x+3 (x-1)

This expression is a bit more complicated, but we can still factor it using some algebraic manipulation. First, we can simplify the left-hand side by squaring both sides of the equation:

4x^2 - 12x + 9 = 44x^2 - 88x + 44

Next, we can combine like terms on both sides of the equation:

-40x^2 + 76x - 35 = 0

At this point, we can use the quadratic formula to solve for x, or we can try to factor the expression. Factoring this expression is a bit tricky, but one possible approach is to use the AC method. This involves finding two numbers that multiply to -35 and add to 76, the coefficients of x in the original equation. One possible pair of numbers is 35 and -1, which gives:

-40x^2 + 76x - 35 = (-5x + 7)(8x - 5)

Therefore, we can write:

2(11√ x²=2x-3 = (x+3 (x-1) = 2(-5x + 7)(8x - 5)

Expression 3: 2 x ² + 2 x - 8 = 4x + 4) (X-2)

This expression can be factored using the grouping method. We can first factor out a 2 from the first two terms:

2x^2 + 2x - 8 = 2(x^2 + x - 4)

Next, we can factor the expression inside the parentheses using the quadratic formula:

x = (-1 ± sqrt(17))/2

Therefore, we can write:

2x^2 + 2x - 8 = 2(x - 2)(x + 2 + sqrt(17))(x + 2 - sqrt(17))

Expression 4: 2 x ² + 3x - 2 = (2x+1) (X-2)

This expression can also be factored using the grouping method. We can first factor out a 2 from the first two terms:

2x^2 + 3x - 2 = 2(x^2 + (3/2)x - 1)

Next, we can factor the expression inside the parentheses using the quadratic formula:

x = (-3 ± sqrt(13))/4

Therefore, we can write:

2x^2 + 3x - 2 = 2(x - (1/2 + sqrt(13)/4))(x - (1/2 - sqrt(13)/4))

In summary, factoring is a useful tool in mathematics that helps simplify expressions and solve equations. In the context provided, we have shown how to factor four expressions using different methods. It is important to note that there are often multiple ways to factor an expression, and the method used may depend on the specific expression.