Factorise : `(x^(2) -3x)(x^(2)-3x-1)-20`

Answer

To factorise the expression \((x^2 - 3x)(x^2 - 3x - 1) - 20\), we will follow these steps:

### Step 1: Substitute \( t \)
Let \( t = x^2 - 3x \). Then, we can rewrite the expression as:
\[ t(t - 1) - 20 \]

### Step 2: Expand the expression
Now, expand the expression:
\[ t(t - 1) - 20 = t^2 - t - 20 \]

### Step 3: Factor the quadratic expression
Next, we need to factor the quadratic \( t^2 - t - 20 \). We look for two numbers that multiply to \(-20\) (the constant term) and add to \(-1\) (the coefficient of \(t\)). The numbers that satisfy this are \(-5\) and \(4\).

So, we can write:
\[ t^2 - t - 20 = (t - 5)(t + 4) \]

### Step 4: Substitute back \( t \)
Now, substitute back \( t = x^2 - 3x \):
\[ (x^2 - 3x - 5)(x^2 - 3x + 4) \]

### Final Answer
Thus, the factorised form of the expression \((x^2 - 3x)(x^2 - 3x - 1) - 20\) is:
\[ (x^2 - 3x - 5)(x^2 - 3x + 4) \]

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