Fractorise the
`a ^(3) - 4a ^(2) + 12 - 3a `

Answer

To factorize the expression \( a^3 - 4a^2 + 12 - 3a \), we can follow these steps:

### Step 1: Rearrange the terms
First, we can rearrange the expression to group similar terms together:
\[ a^3 - 4a^2 - 3a + 12 \]

### Step 2: Group the terms
Next, we can group the terms in pairs:
\[ (a^3 - 4a^2) + (-3a + 12) \]

### Step 3: Factor out the common factors
Now, we can factor out the common factors from each group:
- From the first group \( a^3 - 4a^2 \), we can factor out \( a^2 \):
\[ a^2(a - 4) \]
- From the second group \( -3a + 12 \), we can factor out \(-3\):
\[ -3(a - 4) \]

So, we can rewrite the expression as:
\[ a^2(a - 4) - 3(a - 4) \]

### Step 4: Factor out the common binomial factor
Now, we can see that \( (a - 4) \) is a common factor:
\[ (a - 4)(a^2 - 3) \]

### Step 5: Factor the quadratic expression
The expression \( a^2 - 3 \) can be further factored using the difference of squares:
\[ a^2 - 3 = (a - \sqrt{3})(a + \sqrt{3}) \]

### Final Factorization
Putting it all together, we have:
\[ (a - 4)(a - \sqrt{3})(a + \sqrt{3}) \]

### Final Answer
Thus, the complete factorization of the expression \( a^3 - 4a^2 + 12 - 3a \) is:
\[ (a - 4)(a - \sqrt{3})(a + \sqrt{3}) \]

---