Factorise :
`a(a-1)-b(b-1)`

Answer

To factorise the expression \( a(a-1) - b(b-1) \), we will follow these steps:

### Step 1: Expand the expression
First, we will expand both terms in the expression.

\[ a(a-1) = a^2 - a \]
\[ b(b-1) = b^2 - b \]

So, the expression becomes:

\[ a(a-1) - b(b-1) = (a^2 - a) - (b^2 - b) \]

### Step 2: Simplify the expression
Now, we simplify the expression:

\[ = a^2 - a - b^2 + b \]
\[ = a^2 - b^2 - a + b \]

### Step 3: Rearrange the expression
Next, we rearrange the terms:

\[ = a^2 - b^2 - a + b \]

### Step 4: Factor using the difference of squares
We notice that \( a^2 - b^2 \) is a difference of squares, which can be factored as:

\[ a^2 - b^2 = (a-b)(a+b) \]

So we rewrite the expression:

\[ = (a-b)(a+b) - (a-b) \]

### Step 5: Factor out the common term
Now we can factor out the common term \( (a-b) \):

\[ = (a-b)((a+b) - 1) \]

### Final Result
Thus, the factorised form of the expression \( a(a-1) - b(b-1) \) is:

\[ (a-b)(a+b-1) \]

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