Multiply : `(sqrt(3)+sqrt(2)) (5sqrt(2)+sqrt(3))`

Answer

To solve the problem of multiplying the expressions \((\sqrt{3} + \sqrt{2})(5\sqrt{2} + \sqrt{3})\), we will use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step solution:

### Step 1: Distribute the first term of the first expression
We start by multiplying \(\sqrt{3}\) with each term in the second expression:
\[ \sqrt{3} \cdot 5\sqrt{2} + \sqrt{3} \cdot \sqrt{3} \]
This simplifies to:
\[ 5\sqrt{6} + 3 \]

### Step 2: Distribute the second term of the first expression
Next, we multiply \(\sqrt{2}\) with each term in the second expression:
\[ \sqrt{2} \cdot 5\sqrt{2} + \sqrt{2} \cdot \sqrt{3} \]
This simplifies to:
\[ 5 \cdot 2 + \sqrt{6} = 10 + \sqrt{6} \]

### Step 3: Combine all the terms
Now we combine all the results from Step 1 and Step 2:
\[ (5\sqrt{6} + 3) + (10 + \sqrt{6}) \]
This gives us:
\[ 5\sqrt{6} + \sqrt{6} + 3 + 10 \]

### Step 4: Simplify the expression
Combine like terms:
\[ (5\sqrt{6} + \sqrt{6}) + (3 + 10) = 6\sqrt{6} + 13 \]

### Final Answer
Thus, the final result of the multiplication is:
\[ 6\sqrt{6} + 13 \]

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