Factorise: (i) `4sqrt(3)x^(2) + 5x-2sqrt(3)`
(ii) `7sqrt(2)x^(2)-10x - 4sqrt(2)`

Answer

Let's factorise the given expressions step by step.

### (i) Factorise: \( 4\sqrt{3}x^2 + 5x - 2\sqrt{3} \)

**Step 1: Identify the coefficients**
- The coefficients are \( a = 4\sqrt{3} \), \( b = 5 \), and \( c = -2\sqrt{3} \).

**Step 2: Multiply \( a \) and \( c \)**
- Calculate \( ac = 4\sqrt{3} \times -2\sqrt{3} = -8 \times 3 = -24 \).

**Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)**
- We need two numbers that multiply to \(-24\) and add to \(5\). These numbers are \(8\) and \(-3\).

**Step 4: Rewrite the middle term using these two numbers**
- Rewrite \( 5x \) as \( 8x - 3x \):
\[ 4\sqrt{3}x^2 + 8x - 3x - 2\sqrt{3} \]

**Step 5: Group the terms**
- Group the first two and the last two terms:
\[ (4\sqrt{3}x^2 + 8x) + (-3x - 2\sqrt{3}) \]

**Step 6: Factor out the common factors**
- From the first group, factor out \( 4x \):
\[ 4x(\sqrt{3}x + 2) \]
- From the second group, factor out \(-\sqrt{3}\):
\[ -\sqrt{3}(3x + 2) \]

**Step 7: Combine the factored terms**
- Now we have:
\[ 4x(\sqrt{3}x + 2) - \sqrt{3}(3x + 2) \]
- Notice that \( \sqrt{3}x + 2 \) is common:
\[ (4x - \sqrt{3})(\sqrt{3}x + 2) \]

**Final Factorised Form:**
\[ (4x - \sqrt{3})(\sqrt{3}x + 2) \]