Factorise :
`12 - (x+x^(2)) (8-x-x^(2))`.

Answer

To factorise the expression \( 12 - (x + x^2)(8 - x - x^2) \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the given expression:
\[ 12 - (x + x^2)(8 - x - x^2) \]

### Step 2: Simplify the expression inside the brackets
Let \( u = x + x^2 \). Then, we can rewrite the expression as:
\[ 12 - u(8 - u) \]

### Step 3: Expand the expression
Now, we expand \( u(8 - u) \):
\[ u(8 - u) = 8u - u^2 \]
Thus, the expression becomes:
\[ 12 - (8u - u^2) = 12 - 8u + u^2 \]

### Step 4: Rearrange the expression
Rearranging gives us:
\[ u^2 - 8u + 12 \]

### Step 5: Factor the quadratic expression
Next, we need to factor the quadratic \( u^2 - 8u + 12 \). We look for two numbers that multiply to \( 12 \) and add to \( -8 \). These numbers are \( -6 \) and \( -2 \):
\[ u^2 - 8u + 12 = (u - 6)(u - 2) \]

### Step 6: Substitute back for \( u \)
Now we substitute back \( u = x + x^2 \):
\[ (x + x^2 - 6)(x + x^2 - 2) \]

### Step 7: Factor each expression
We can factor each expression further if possible. Let's analyze:
1. \( x + x^2 - 6 = x^2 + x - 6 \)
2. \( x + x^2 - 2 = x^2 + x - 2 \)

### Step 8: Factor \( x^2 + x - 6 \)
To factor \( x^2 + x - 6 \), we look for two numbers that multiply to \( -6 \) and add to \( 1 \). These numbers are \( 3 \) and \( -2 \):
\[ x^2 + x - 6 = (x - 2)(x + 3) \]

### Step 9: Factor \( x^2 + x - 2 \)
To factor \( x^2 + x - 2 \), we look for two numbers that multiply to \( -2 \) and add to \( 1 \). These numbers are \( 2 \) and \( -1 \):
\[ x^2 + x - 2 = (x - 1)(x + 2) \]

### Final Step: Combine the factors
Thus, the complete factorization of the original expression is:
\[ (x - 2)(x + 3)(x - 1)(x + 2) \]

### Summary
The final factorized form of the expression \( 12 - (x + x^2)(8 - x - x^2) \) is:
\[ (x - 2)(x + 3)(x - 1)(x + 2) \]