Question

Factorise the following:

(2x² – x) (2x² – x – 4) + 3

Answer 1

Given: (2x² – x) (2x² – x – 4) + 3

Step-wise calculation:

Step 1: Use substitution for simplification

Observe the repeated expression (2x² – x).

Let y = 2x² – x.

The original expression becomes y(y – 4) + 3.

Step 2: Factor the simplified quadratic expression

 Expand and factor the quadratic expression in y:

y² – 4y + 3

Find two numbers that multiply to 3 and sum to –4 which are –1 and –3:

(y – 1)(y – 3)

Step 3: Substitute the original terms back

Replace y with 2x² – x:

(2x² – x – 1)(2x² – x – 3)

Step 4: Factor the resulting quadratic expressions

Factor each of the two new quadratic expressions by splitting the middle term:

For the first factor (2x² – x – 1):

2x² – 2x + x – 1

2x(x – 1) + 1(x – 1)

(2x + 1)(x – 1)

For the first factor (2x² – x – 3):

2x² – 3x + 2x – 3

x(2x – 3) + 1(2x – 3)

(x + 1)(2x – 3)

Step 5: Combine all factors

Combine all the linear factors into the final factored form:

(2x + 1)(x – 1)(x + 1)(2x – 3)