Factorise :
`4x^(4) - x^(2) - 12x - 36`

Answer

To factorise the expression \( 4x^4 - x^2 - 12x - 36 \), we will follow these steps:

### Step 1: Group the terms
We start with the expression:
\[ 4x^4 - x^2 - 12x - 36 \]
We can rearrange it as:
\[ 4x^4 - x^2 - (12x + 36) \]

### Step 2: Factor out common terms
Next, we can factor out \( -1 \) from the last two terms:
\[ 4x^4 - x^2 - 12(x + 3) \]

### Step 3: Rearrange and group
Now, we can group the first two terms and the last two terms:
\[ (4x^4 - x^2) - (12x + 36) \]

### Step 4: Factor out common factors
From the first group \( 4x^4 - x^2 \), we can factor out \( x^2 \):
\[ x^2(4x^2 - 1) - 12(x + 3) \]

### Step 5: Recognize the difference of squares
Notice that \( 4x^2 - 1 \) can be factored as a difference of squares:
\[ 4x^2 - 1 = (2x - 1)(2x + 1) \]
Thus, we rewrite the expression:
\[ x^2(2x - 1)(2x + 1) - 12(x + 3) \]

### Step 6: Factor by grouping
Now we can factor by grouping:
\[ x^2(2x - 1)(2x + 1) - 12(x + 3) = (2x - 1)(2x + 1)(x^2 - 12) \]

### Step 7: Factor the quadratic
The term \( x^2 - 12 \) can be factored further:
\[ x^2 - 12 = (x - 2\sqrt{3})(x + 2\sqrt{3}) \]

### Final Factorization
Putting it all together, we have:
\[ (2x - 1)(2x + 1)(x - 2\sqrt{3})(x + 2\sqrt{3}) \]

### Summary of the Factorization
The complete factorization of \( 4x^4 - x^2 - 12x - 36 \) is:
\[ (2x - 1)(2x + 1)(x - 2\sqrt{3})(x + 2\sqrt{3}) \]