Factorise : (ii) `7+10 (x-y) - 8 (x-y)^2`

Answer

To factorise the expression \( 7 + 10(x - y) - 8(x - y)^2 \), we can follow these steps:

### Step 1: Substitute \( x - y \)
Let \( z = x - y \). Then, we can rewrite the expression as:
\[ 7 + 10z - 8z^2 \]

### Step 2: Rearrange the expression
Rearranging the expression gives us:
\[ -8z^2 + 10z + 7 \]
To make it easier to factor, we can multiply through by -1:
\[ 8z^2 - 10z - 7 \]

### Step 3: Factor the quadratic expression
We want to factor \( 8z^2 - 10z - 7 \). We need two numbers that multiply to \( 8 \times -7 = -56 \) and add to \( -10 \). The numbers that work are \( -14 \) and \( 4 \).

### Step 4: Rewrite the middle term
We can rewrite the expression as:
\[ 8z^2 - 14z + 4z - 7 \]

### Step 5: Factor by grouping
Now, we group the terms:
\[ (8z^2 - 14z) + (4z - 7) \]
Factoring out the common factors in each group gives us:
\[ 2z(4z - 7) + 1(4z - 7) \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \( (4z - 7) \):
\[ (4z - 7)(2z + 1) \]

### Step 7: Substitute back \( z = x - y \)
Now we substitute back \( z = x - y \):
\[ (4(x - y) - 7)(2(x - y) + 1) \]
This simplifies to:
\[ (4x - 4y - 7)(2x - 2y + 1) \]

### Final Answer
Thus, the factorised form of the expression \( 7 + 10(x - y) - 8(x - y)^2 \) is:
\[ (4x - 4y - 7)(2x - 2y + 1) \]