Factorise :
`7-12x - 4x^(2)`

Answer

To factorise the expression \( 7 - 12x - 4x^2 \), we will follow these steps:

### Step 1: Rearrange the expression
We can rewrite the expression in standard form, which is usually in descending order of powers of \( x \):
\[ -4x^2 - 12x + 7 \]

### Step 2: Identify coefficients
In the expression \(-4x^2 - 12x + 7\), we identify:
- \( a = -4 \) (coefficient of \( x^2 \))
- \( b = -12 \) (coefficient of \( x \))
- \( c = 7 \) (constant term)

### Step 3: Calculate the product \( ac \)
Next, we calculate the product \( ac \):
\[ ac = (-4) \times 7 = -28 \]

### Step 4: Split the middle term
We need to find two numbers that multiply to \( ac = -28 \) and add to \( b = -12 \). The numbers that satisfy this condition are \( -14 \) and \( 2 \) because:
\[ -14 + 2 = -12 \quad \text{and} \quad -14 \times 2 = -28 \]

### Step 5: Rewrite the expression
Now, we can rewrite the expression by splitting the middle term:
\[ -4x^2 - 14x + 2x + 7 \]

### Step 6: Group the terms
Next, we group the terms:
\[ (-4x^2 - 14x) + (2x + 7) \]

### Step 7: Factor out the common terms
Now, we factor out the common factors from each group:
\[ -2x(2x + 7) + 1(2x + 7) \]

### Step 8: Factor by grouping
Now we can factor out the common binomial factor \( (2x + 7) \):
\[ (2x + 7)(-2x + 1) \]

### Step 9: Write the final factorised form
Thus, the factorised form of the expression \( 7 - 12x - 4x^2 \) is:
\[ (2x + 7)(1 - 2x) \]