Explanation
To evaluate the given expression, we can use the trigonometric identity $\\sin (A+B) = \\sin A \\cos B + \\cos A \\sin B$. We are given $A = 60^\\circ$ and $B = 30^\\circ$. Substituting these values in the expression, we get $2 \\sqrt{2} \\cos 45^\\circ \\sin 30^\\circ + 2 \\sqrt{3} \\cos 30^\\circ$. Using the trigonometric values of $\\cos 45^\\circ = \\frac{1}{\\sqrt{2}}$, $\\sin 30^\\circ = \\frac{1}{2}$, and $\\cos 30^\\circ = \\frac{\\sqrt{3}}{2}$, we can simplify the expression to $2 \\sqrt{2} \\cdot \\frac{1}{\\sqrt{2}} \\cdot \\frac{1}{2} + 2 \\sqrt{3} \\cdot \\frac{\\sqrt{3}}{2} = 1 + 3 = 4$.
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