Let's break down and expand each of the given expressions individually using the distributive property.
### 1. Expanding [tex]\(u(x-y)^2\)[/tex]:
First, recall the algebraic identity for the square of a binomial:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
Here, we have [tex]\((x - y)^2\)[/tex], so we substitute [tex]\(a = x\)[/tex] and [tex]\(b = y\)[/tex]:
[tex]\[
(x - y)^2 = x^2 - 2xy + y^2
\][/tex]
Now, we multiply the entire expression by [tex]\(u\)[/tex]:
[tex]\[
u(x - y)^2 = u \cdot (x^2 - 2xy + y^2) = ux^2 - 2uxy + uy^2
\][/tex]
### 2. Expanding [tex]\(6(x^2 - 4)\)[/tex]:
The expression contains a monomial factor multiplying a binomial. According to the distributive property:
[tex]\[
a(b + c) = ab + ac
\][/tex]
Here, we have:
[tex]\[
6(x^2 - 4) = 6 \cdot x^2 - 6 \cdot 4 = 6x^2 - 24
\][/tex]
### 3. Expanding [tex]\(8(x+y)^2\)[/tex]:
Again, we recall the algebraic identity for the square of a binomial:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
Here, we have [tex]\((x + y)^2\)[/tex], so we substitute [tex]\(a = x\)[/tex] and [tex]\(b = y\)[/tex]:
[tex]\[
(x + y)^2 = x^2 + 2xy + y^2
\][/tex]
Now, we multiply the entire expression by [tex]\(8\)[/tex]:
[tex]\[
8(x + y)^2 = 8 \cdot (x^2 + 2xy + y^2) = 8x^2 + 16xy + 8y^2
\][/tex]
### Summary
The expanded forms of the given expressions are:
1. [tex]\( u(x-y)^2 = ux^2 - 2uxy + uy^2 \)[/tex]
2. [tex]\( 6(x^2 - 4) = 6x^2 - 24 \)[/tex]
3. [tex]\( 8(x+y)^2 = 8x^2 + 16xy + 8y^2 \)[/tex]
These are the desired results.
