To find the product [tex]\((2 \sqrt{7} + 3 \sqrt{6})(5 \sqrt{2} + 4 \sqrt{3})\)[/tex], we need to distribute each term in the first parentheses to each term in the second parentheses using the distributive property.
1. Multiply [tex]\(2 \sqrt{7}\)[/tex] by each term in [tex]\(5 \sqrt{2} + 4 \sqrt{3}\)[/tex]:
[tex]\[
2 \sqrt{7} \cdot 5 \sqrt{2} + 2 \sqrt{7} \cdot 4 \sqrt{3}
\][/tex]
This gives us:
[tex]\[
2 \cdot 5 \cdot \sqrt{7 \cdot 2} + 2 \cdot 4 \cdot \sqrt{7 \cdot 3}
\][/tex]
[tex]\[
10 \sqrt{14} + 8 \sqrt{21}
\][/tex]
2. Similarly, multiply [tex]\(3 \sqrt{6}\)[/tex] by each term in [tex]\(5 \sqrt{2} + 4 \sqrt{3}\)[/tex]:
[tex]\[
3 \sqrt{6} \cdot 5 \sqrt{2} + 3 \sqrt{6} \cdot 4 \sqrt{3}
\][/tex]
This gives us:
[tex]\[
3 \cdot 5 \cdot \sqrt{6 \cdot 2} + 3 \cdot 4 \cdot \sqrt{6 \cdot 3}
\][/tex]
[tex]\[
15 \sqrt{12} + 12 \sqrt{18}
\][/tex]
3. Now combine all the terms:
[tex]\[
10 \sqrt{14} + 8 \sqrt{21} + 15 \sqrt{12} + 12 \sqrt{18}
\][/tex]
4. Simplify the terms:
[tex]\[
15 \sqrt{12} = 15 \sqrt{4 \cdot 3} = 15 \cdot 2 \cdot \sqrt{3} = 30 \sqrt{3}
\][/tex]
[tex]\[
12 \sqrt{18} = 12 \sqrt{9 \cdot 2} = 12 \cdot 3 \cdot \sqrt{2} = 36 \sqrt{2}
\][/tex]
5. Substituting these simplified terms back, we get:
[tex]\[
10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 36 \sqrt{2}
\][/tex]
Hence, the product [tex]\((2 \sqrt{7} + 3 \sqrt{6})(5 \sqrt{2} + 4 \sqrt{3})\)[/tex] results in:
[tex]\[
10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 36 \sqrt{2}
\][/tex]
This matches the following answer from your choices:
[tex]\[
\boxed{10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 36 \sqrt{2}}
\][/tex]
