Let's break down each of these expressions step-by-step and solve them:
### 6. [tex]\((x-4)(x+4)\)[/tex]
This is a classic difference of squares problem. The formula for the difference of squares is:
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
In this case, [tex]\(a = x\)[/tex] and [tex]\(b = 4\)[/tex]. Applying the formula, we get:
[tex]\[
(x - 4)(x + 4) = x^2 - 4^2 = x^2 - 16
\][/tex]
So, the expanded form is:
[tex]\[
x^2 - 16
\][/tex]
### 7. [tex]\((2r-5)(2r-5)\)[/tex]
This problem involves squaring a binomial. The formula for the square of a binomial is:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
Here, [tex]\(a = 2r\)[/tex] and [tex]\(b = 5\)[/tex]. Applying the formula, we get:
[tex]\[
(2r - 5)^2 = (2r)^2 - 2 \cdot 2r \cdot 5 + 5^2 = 4r^2 - 20r + 25
\][/tex]
So, the expanded form is:
[tex]\[
4r^2 - 20r + 25
\][/tex]
### x. [tex]\((3 - 4m^2)\)[/tex]
This expression is already in its simplest form, and no expansion is needed. The expression remains:
[tex]\[
3 - 4m^2
\][/tex]
### 9. [tex]\((2h+7)(2h-7)\)[/tex]
Again, this is a difference of squares problem. Using the formula:
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
We have [tex]\(a = 2h\)[/tex] and [tex]\(b = 7\)[/tex]. Applying the formula, we get:
[tex]\[
(2h + 7)(2h - 7) = (2h)^2 - 7^2 = 4h^2 - 49
\][/tex]
So, the expanded form is:
[tex]\[
4h^2 - 49
\][/tex]
### 10. [tex]\((8 - 3x)(8 + 3y)\)[/tex]
This expression requires a straightforward distribution (also known as the FOIL method). Multiply each term in the first binomial by each term in the second binomial:
[tex]\[
(8 - 3x)(8 + 3y) = 8 \cdot 8 + 8 \cdot 3y - 3x \cdot 8 - 3x \cdot 3y
\][/tex]
Simplifying each product, we get:
[tex]\[
= 64 + 24y - 24x - 9xy
\][/tex]
So, the expanded form is:
[tex]\[
64 + 24y - 24x - 9xy
\][/tex]
### Summary of Solutions
[tex]\[
\begin{array}{l}
6. \quad (x - 4)(x + 4) = x^2 - 16 \\
7. \quad (2r - 5)(2r - 5) = 4r^2 - 20r + 25 \\
x. \quad 3 - 4m^2 \\
9. \quad (2h + 7)(2h - 7) = 4h^2 - 49 \\
10. \quad (8 - 3x)(8 + 3y) = 64 + 24y - 24x - 9xy \\
\end{array}
\][/tex]
These are the detailed, step-by-step solutions to the given problems!
