Let's solve the problem step by step.
We need to multiply the binomials [tex]\((\sqrt{10} + 2\sqrt{8})(\sqrt{10} - 2\sqrt{8})\)[/tex].
This expression takes the form of a difference of squares, which allows us to use the formula:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
Here, we have:
[tex]\[
a = \sqrt{10}
\][/tex]
[tex]\[
b = 2\sqrt{8}
\][/tex]
First, let's calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = (\sqrt{10})^2 = 10
\][/tex]
Next, let's calculate [tex]\(b^2\)[/tex]:
[tex]\[
b = 2\sqrt{8} \rightarrow b^2 = (2\sqrt{8})^2 = 4 \times 8 = 32
\][/tex]
Using the difference of squares formula, we now compute:
[tex]\[
(\sqrt{10} + 2\sqrt{8})(\sqrt{10} - 2\sqrt{8}) = a^2 - b^2 = 10 - 32 = -22
\][/tex]
Thus, the result of the multiplication is:
[tex]\[
-22
\][/tex]
