Factorise :
`3 - 5x + 5y - 12(x-y)^(2)`

Answer

To factorise the expression \(3 - 5x + 5y - 12(x-y)^2\), we will follow these steps:

### Step 1: Rewrite the expression
Start by rewriting the expression clearly:
\[ 3 - 5x + 5y - 12(x-y)^2 \]

### Step 2: Expand the square
Next, expand the term \(-12(x-y)^2\):
\[ (x-y)^2 = x^2 - 2xy + y^2 \]
Thus,
\[ -12(x-y)^2 = -12(x^2 - 2xy + y^2) = -12x^2 + 24xy - 12y^2 \]

### Step 3: Substitute back into the expression
Now substitute this back into the expression:
\[ 3 - 5x + 5y - 12x^2 + 24xy - 12y^2 \]

### Step 4: Rearrange the expression
Rearranging the expression gives:
\[ -12x^2 + 24xy - 12y^2 - 5x + 5y + 3 \]

### Step 5: Group the terms
Group the quadratic terms and the linear terms:
\[ -12(x^2 - 2xy + y^2) - 5x + 5y + 3 \]

### Step 6: Factor out common terms
Notice that we can factor out \(-12\) from the quadratic part:
\[ -12((x-y)^2) - 5(x - y) + 3 \]

### Step 7: Let \(a = x - y\)
Let \(a = x - y\):
\[ -12a^2 - 5a + 3 \]

### Step 8: Factor the quadratic
Now, we need to factor the quadratic \(-12a^2 - 5a + 3\). We can use the method of splitting the middle term:
- The product of the coefficient of \(a^2\) and the constant term: \(-12 \times 3 = -36\)
- We need two numbers that multiply to \(-36\) and add to \(-5\). The numbers are \(-9\) and \(4\).

### Step 9: Rewrite the quadratic
Rewrite the quadratic:
\[ -12a^2 - 9a + 4a + 3 \]

### Step 10: Factor by grouping
Group the terms:
\[ (-12a^2 - 9a) + (4a + 3) \]
Factor out common terms:
\[ -3a(4a + 3) + 1(4a + 3) \]
Now factor out \((4a + 3)\):
\[ (4a + 3)(-3a + 1) \]

### Step 11: Substitute back for \(a\)
Substitute back \(a = x - y\):
\[ (4(x - y) + 3)(-3(x - y) + 1) \]

### Final Answer
Thus, the factorised form of the expression is:
\[ (4(x - y) + 3)(1 - 3(x - y)) \]