Factorise : `( 5x -3) ^(2) - ( 5x -3) - 20`

Options

A. `(5x + 8) ( 5x -1)`
B. `(5x - 8) ( 5x +1)`
C. `(5x - 8) ( 5x -1)`
D. `(5x + 8) (5x +1)`

Answer

To factorise the expression \( (5x - 3)^2 - (5x - 3) - 20 \), we can follow these steps:

### Step 1: Substitute
Let \( t = 5x - 3 \). Then, the expression becomes:
\[ t^2 - t - 20 \]

### Step 2: Factor the Quadratic
Now, we need to factor the quadratic expression \( t^2 - t - 20 \). We are looking for two numbers that multiply to \(-20\) (the constant term) and add up to \(-1\) (the coefficient of \(t\)).

The numbers that satisfy this condition are \( -5 \) and \( 4 \) because:
\[ -5 \times 4 = -20 \quad \text{and} \quad -5 + 4 = -1 \]

Thus, we can factor the quadratic as:
\[ (t - 5)(t + 4) \]

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

This simplifies to:
\[ (5x - 8)(5x + 1) \]

### Final Answer
Therefore, the factorised form of the expression \( (5x - 3)^2 - (5x - 3) - 20 \) is:
\[ (5x - 8)(5x + 1) \]

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