Foctorise :
` 7sqrt2 x^(2) - 10 x - 4sqrt2`

Answer

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

### Step 1: Identify the coefficients
The expression is in the standard quadratic 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 \( a \) and \( c \):
\[ a \cdot c = 7\sqrt{2} \cdot (-4\sqrt{2}) = -28 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \(-28\) and add to \(-10\). The numbers are \(-14\) and \(4\):
\[ -14 + 4 = -10 \quad \text{and} \quad -14 \cdot 4 = -56 \]

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

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

### Step 6: Factor out the common terms from each group
From the first group \( 7\sqrt{2} x^2 - 14x \), we can factor out \( 7\sqrt{2} x \):
\[ 7\sqrt{2} x (x - 2) \]
From the second group \( 4x - 4\sqrt{2} \), we can factor out \( 4 \):
\[ 4(x - \sqrt{2}) \]

### Step 7: Combine the factored terms
Now we can combine the factored terms:
\[ 7\sqrt{2} x (x - 2) + 4(x - \sqrt{2}) \]
Notice that both groups contain a common factor \( (x - 2) \):
\[ (x - 2)(7\sqrt{2} x + 4) \]

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

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