To find the product of the expression [tex]\((2\sqrt{7} + 3\sqrt{6})(5\sqrt{2} + 4\sqrt{3})\)[/tex], follow these detailed steps:
1. Start by expanding the product using the distributive property (FOIL method for binomials):
[tex]\[
(2\sqrt{7} + 3\sqrt{6})(5\sqrt{2} + 4\sqrt{3}) = 2\sqrt{7} \cdot 5\sqrt{2} + 2\sqrt{7} \cdot 4\sqrt{3} + 3\sqrt{6} \cdot 5\sqrt{2} + 3\sqrt{6} \cdot 4\sqrt{3}
\][/tex]
2. Calculate each multiplication individually:
- For the first term:
[tex]\[
2\sqrt{7} \cdot 5\sqrt{2} = 2 \cdot 5 \cdot \sqrt{7 \cdot 2} = 10\sqrt{14}
\][/tex]
- For the second term:
[tex]\[
2\sqrt{7} \cdot 4\sqrt{3} = 2 \cdot 4 \cdot \sqrt{7 \cdot 3} = 8\sqrt{21}
\][/tex]
- For the third term:
[tex]\[
3\sqrt{6} \cdot 5\sqrt{2} = 3 \cdot 5 \cdot \sqrt{6 \cdot 2} = 15\sqrt{12}
\][/tex]
Simplify [tex]\(\sqrt{12}\)[/tex] to get:
[tex]\[
15\sqrt{12} = 15 \cdot 2\sqrt{3} = 30\sqrt{3}
\][/tex]
- For the fourth term:
[tex]\[
3\sqrt{6} \cdot 4\sqrt{3} = 3 \cdot 4 \cdot \sqrt{6 \cdot 3} = 12\sqrt{18}
\][/tex]
Simplify [tex]\(\sqrt{18}\)[/tex] to get:
[tex]\[
12\sqrt{18} = 12 \cdot 3\sqrt{2} = 36\sqrt{2}
\][/tex]
3. Combine all terms to form the expanded expression:
[tex]\[
(2\sqrt{7} + 3\sqrt{6})(5\sqrt{2} + 4\sqrt{3}) = 10\sqrt{14} + 8\sqrt{21} + 30\sqrt{3} + 36\sqrt{2}
\][/tex]
4. The expanded product of the given expression is:
[tex]\[
10\sqrt{14} + 8\sqrt{21} + 30\sqrt{3} + 36\sqrt{2}
\][/tex]
This matches the fourth provided option:
[tex]\[
10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 36 \sqrt{2}
\][/tex]
Thus, the correct answer is:
[tex]\[
10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 36 \sqrt{2}
\][/tex]
