Certainly! Let's multiply the two binomials [tex]\( (8 \sqrt{2} + 2 \sqrt{10})(\sqrt{10} - 8 \sqrt{2}) \)[/tex] step-by-step, using the distributive property, specifically the FOIL method which stands for First, Outer, Inner, Last.
### Step 1: Multiply the First Terms
The first terms are [tex]\( 8 \sqrt{2} \)[/tex] and [tex]\( \sqrt{10} \)[/tex].
[tex]\[ 8 \sqrt{2} \cdot \sqrt{10} = 8 \sqrt{2} \cdot \sqrt{10} = 8 \sqrt{20} = 8 \cdot 2 \sqrt{5} = 16 \sqrt{5} \][/tex]
### Step 2: Multiply the Outer Terms
The outer terms are [tex]\( 8 \sqrt{2} \)[/tex] and [tex]\( -8 \sqrt{2} \)[/tex].
[tex]\[ 8 \sqrt{2} \cdot -8 \sqrt{2} = 8 \cdot -8 \cdot \sqrt{2} \cdot \sqrt{2} = -64 \cdot 2 = -128 \][/tex]
### Step 3: Multiply the Inner Terms
The inner terms are [tex]\( 2 \sqrt{10} \)[/tex] and [tex]\( \sqrt{10} \)[/tex].
[tex]\[ 2 \sqrt{10} \cdot \sqrt{10} = 2 \cdot 10 = 20 \][/tex]
### Step 4: Multiply the Last Terms
The last terms are [tex]\( 2 \sqrt{10} \)[/tex] and [tex]\( -8 \sqrt{2} \)[/tex].
[tex]\[ 2 \sqrt{10} \cdot -8 \sqrt{2} = 2 \cdot -8 \cdot \sqrt{10} \cdot \sqrt{2} = -16 \sqrt{20} = -16 \cdot 2 \sqrt{5} = -32 \sqrt{5} \][/tex]
### Step 5: Combine All the Terms
Now we add all these results together:
[tex]\[ (16 \sqrt{5}) + (-128) + (20) + (-32 \sqrt{5}) \][/tex]
Combine like terms:
[tex]\[ 16 \sqrt{5} - 32 \sqrt{5} = -16 \sqrt{5} \][/tex]
[tex]\[ -128 + 20 = -108 \][/tex]
So the resulting simplified expression is:
[tex]\[ \boxed{-16 \sqrt{5} - 108} \][/tex]
