Factorise:` 2 sqrt2 x ^(3) + 3 sqrt3 y ^(3) + sqrt5 (5 -3 sqrt6 xy)`

Answer

To factorize the expression \( 2 \sqrt{2} x^3 + 3 \sqrt{3} y^3 + \sqrt{5} (5 - 3 \sqrt{6} xy) \), we will follow these steps:

### Step 1: Distribute the \(\sqrt{5}\)
First, we need to distribute \(\sqrt{5}\) in the expression:
\[ \sqrt{5} (5 - 3 \sqrt{6} xy) = 5\sqrt{5} - 3\sqrt{30} xy \]
Now, we can rewrite the expression:
\[ 2 \sqrt{2} x^3 + 3 \sqrt{3} y^3 + 5\sqrt{5} - 3\sqrt{30} xy \]

### Step 2: Rearrange the terms
Next, let's rearrange the terms for clarity:
\[ 2 \sqrt{2} x^3 - 3\sqrt{30} xy + 3 \sqrt{3} y^3 + 5\sqrt{5} \]

### Step 3: Group the terms
Now, we can group the terms in a way that might reveal a common factor:
\[ (2 \sqrt{2} x^3 + 3 \sqrt{3} y^3) + (5\sqrt{5} - 3\sqrt{30} xy) \]

### Step 4: Factor by grouping
We can factor out common terms from the grouped expressions. The first group can be factored as:
\[ \sqrt{2} (2x^3) + \sqrt{3} (3y^3) = \sqrt{6} (x^3 + y^3) \]
The second group can be rewritten as:
\[ 5\sqrt{5} - 3\sqrt{30} xy = \sqrt{5} (5 - 3\sqrt{6} xy) \]

### Step 5: Combine the factors
Now, we can combine the factored terms:
\[ \sqrt{2} (2x^3 + 3\sqrt{3} y^3) + \sqrt{5} (5 - 3\sqrt{6} xy) \]

### Step 6: Final factorization
The expression can be factored as:
\[ \sqrt{2}(x^3 + y^3) + \sqrt{5}(5 - 3\sqrt{6}xy) \]

This is the final factorized form of the expression.