Factorise:
` sqrt5 x^(2) + 2x- 3sqrt5`

Answer

To factorise the expression \( \sqrt{5} x^2 + 2x - 3\sqrt{5} \), we can follow these steps:

### Step 1: Identify coefficients
The expression is in the form \( ax^2 + bx + c \), where:
- \( a = \sqrt{5} \)
- \( b = 2 \)
- \( c = -3\sqrt{5} \)

### Step 2: Calculate the product of \( a \) and \( c \)
We need to find the product \( ac \):
\[ ac = \sqrt{5} \times (-3\sqrt{5}) = -3 \times 5 = -15 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \(-15\) (the product we calculated) and add up to \(2\) (the coefficient \(b\)). The numbers \(5\) and \(-3\) satisfy this condition:
\[ 5 \times (-3) = -15 \quad \text{and} \quad 5 + (-3) = 2 \]

### Step 4: Rewrite the middle term
Using the numbers found, we can rewrite the expression:
\[ \sqrt{5} x^2 + 5x - 3x - 3\sqrt{5} \]

### Step 5: Group the terms
Now, we group the terms:
\[ (\sqrt{5} x^2 + 5x) + (-3x - 3\sqrt{5}) \]

### Step 6: Factor out common terms from each group
From the first group \( \sqrt{5} x^2 + 5x \), we can factor out \( x \):
\[ x(\sqrt{5} x + 5) \]
From the second group \(-3x - 3\sqrt{5}\), we can factor out \(-3\):
\[ -3(x + \sqrt{5}) \]

### Step 7: Combine the factored terms
Now we can combine the factored terms:
\[ x(\sqrt{5} x + 5) - 3(x + \sqrt{5}) \]
Notice that we can factor out \( (x + \sqrt{5}) \):
\[ (x + \sqrt{5})(\sqrt{5} x - 3) \]

### Final Answer
Thus, the factorised form of the expression \( \sqrt{5} x^2 + 2x - 3\sqrt{5} \) is:
\[ (x + \sqrt{5})(\sqrt{5} x - 3) \]