Please assist me to factorise the following.

4x^3 + 5x^2y - 4xy^2 - 5y^2

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Here is a start.

4x^3 - 4xy^2 + 5x^2y - 5y^2 =

4x(x^2-y^2) + 5y(x^2-y)

To factorize the expression 4x^3 + 5x^2y - 4xy^2 - 5y^2, we can look for common factors among the terms and then group them together.

Step 1: Find the common factors among the terms:
In this case, the common factor is "x" in the first two terms and "-y" in the last two terms.

Step 2: Group the terms:
4x^3 + 5x^2y - 4xy^2 - 5y^2 can be grouped as follows:
(x common factor) (4x^2 - 5y) + (-y common factor) (4xy + 5y)

Step 3: Factor out the common factors:
Taking out the common factors, we get:
x(4x^2 - 5y) - y(4xy + 5y)

So, the fully factorized expression is:
(x - y)(4x^2 - 5y)

To factorize the expression 4x^3 + 5x^2y - 4xy^2 - 5y^2, we will first look for any common factors among the terms.

The terms in the expression have different variables (x and y) and different exponents on those variables. However, there is a common factor in all the terms, which is -1. So, we can factor out -1.

By factoring out -1, we have:

-(4x^3 + 5x^2y - 4xy^2 - 5y^2)

Now, let's focus on the remaining expression inside the parentheses:

4x^3 + 5x^2y - 4xy^2 - 5y^2

Next, we can look for any common factors among the terms again. This time, we can see that all the terms have an x in common.

By factoring out x, we have:

x(4x^2 + 5xy - 4y^2) - 5y^2

Now, we can look at the remaining expression inside the parentheses:

4x^2 + 5xy - 4y^2

At this point, we have a quadratic expression that can be factored further.

To factor the quadratic expression, we need to find two binomials whose product equals the quadratic expression. The form of the binomials is (mx + ny)(px + qy), where m, n, p, and q are coefficients.

To do this, we need to find two numbers whose product is equal to the product of the leading coefficient (4) and the constant term (-4y^2). In this case, the product is -16y^2.

We also need to find two numbers whose sum is equal to the coefficient of the middle term (5xy). In this case, the sum is 5xy.

After some trial and error or using factoring techniques, we can determine that the binomials are:

(2x - y)(2x + 4y)

Therefore, we have factored the expression 4x^3 + 5x^2y - 4xy^2 - 5y^2 as:

-(x)(2x - y)(2x + 4y)