Factorise : `7sqrt(2)x^(2) - 10 x - 4sqrt(2)`

Answer

To factorise the expression \( 7\sqrt{2}x^2 - 10x - 4\sqrt{2} \), we will follow the steps outlined below:

### Step 1: Identify the coefficients
The given quadratic expression is in the standard form \( ax^2 + bx + c \), where:
- \( a = 7\sqrt{2} \)
- \( b = -10 \)
- \( c = -4\sqrt{2} \)

### Step 2: Multiply \( a \) and \( c \)
We need to multiply the coefficient \( a \) and the constant term \( c \):
\[ a \cdot c = 7\sqrt{2} \cdot (-4\sqrt{2}) = -28 \cdot 2 = -56 \]

### Step 3: Find two numbers that multiply to \( -56 \) and add to \( -10 \)
We need to find two numbers that multiply to \( -56 \) and add to \( -10 \). The numbers that satisfy these conditions are \( -14 \) and \( 4 \):
\[ -14 \cdot 4 = -56 \quad \text{and} \quad -14 + 4 = -10 \]

### Step 4: Rewrite the middle term
Now, we can rewrite the expression by splitting the middle term using the numbers we found:
\[ 7\sqrt{2}x^2 - 14x + 4x - 4\sqrt{2} \]

### Step 5: Group the terms
Next, we group the terms:
\[ (7\sqrt{2}x^2 - 14x) + (4x - 4\sqrt{2}) \]

### Step 6: Factor out the common factors
Now, we factor out the common factors from each group:
\[ 7\sqrt{2}x(x - 2) + 4(x - \sqrt{2}) \]

### Step 7: Factor by grouping
We can now factor by grouping:
\[ (7\sqrt{2}x + 4)(x - 2) \]

### Final Factorised Form
Thus, the factorised form of the expression \( 7\sqrt{2}x^2 - 10x - 4\sqrt{2} \) is:
\[ (7\sqrt{2}x + 4)(x - 2) \]