Method 1
Following the "hence" (using what was done before), let $a = x^{2} + 4$ and $b = x^{2} - 2$

$a^{2} - b^{2}$

[1]

Use the "difference of two squares" to factorise

$(a + b) (a - b)$

Replace $a$ with $x^{2} + 4$ in brackets and $b$ with $x^{2} - 2$ in brackets

$((x^{2} + 4) + (x^{2} - 2)) ((x^{2} + 4) - (x^{2} - 2))$

Expand inside the brackets

$(x^{2} + 4 + x^{2} - 2) (x^{2} + 4 - x^{2} + 2)$

[1]

Collect "like" terms

$(2 x^{2} + 2) (6)$

Multiply out the brackets (it may help to write the 6 at the front)

$12 x^{2} + 12$ [1]

Method 2
Following the "or otherwise", write out the squared brackets as the product of two brackets

$(x^{2} + 4) (x^{2} + 4) - (x^{2} - 2) (x^{2} - 2)$

Multiply out the two sets of brackets and collect like terms (keeping the results in their own separate brackets)

$(x^{4} + 4 x^{2} + 4 x^{2} + 16) - (x^{4} - 2 x^{2} - 2 x^{2} + 4) = (x^{4} + 8 x^{2} + 16) - (x^{4} - 4 x^{2} + 4)$

[1]

Remove the brackets (by multiplying all terms by -1 in the second bracket)

$x^{4} + 8 x^{2} + 16 - x^{4} + 4 x^{2} - 4$

[1]

Collect like terms

$12 x^{2} + 12$ [1]

Factorising (OCR GCSE Maths)

Exam Questions

1. Number Operations & Integers

Number Toolkit

Prime Factors, HCF & LCM

2. Fractions, Decimals & Percentages

Fractions

Percentages

Working with FDP

3. Indices & Surds

Powers, Roots & Standard Form

Surds

4. Approximation & Estimation

Rounding, Estimation & Bounds

Using a Calculator

5. Ratio, Proportion & Rates of Change

Ratio Toolkit

Ratio Problem Solving

Direct & Inverse Proportion

Exchange Rates & Best Buys

Simple & Compound Interest, Growth & Decay

6. Algebra

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Algebraic Roots & Indices

Expanding Brackets

Factorising

Completing the Square

Algebraic Fractions

Rearranging Formulae

Algebraic Proof

Linear Equations

Solving Quadratic Equations

Solving Inequalities

Simultaneous Equations

Iteration

Forming & Solving Equations

Functions

Sequences

7. Graphs of Equations & Functions

Coordinate Geometry

Linear Graphs y = mx + c

Graphing Inequalities

Graphs of Functions

Equation of a Circle

Estimating Gradients & Areas under Graphs

Real-Life Graphs

Transformations of Graphs

8. Basic Geometry

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Angles in Polygons & Parallel Lines

Circle Theorems

9. Congruence & Similarity

Congruence, Similarity & Geometrical Proof

Area & Volume of Similar Shapes

Vectors

Transformations

10. Mensuration

Standard & Compound Units

Bearings, Scale Drawing, Constructions & Loci

Area & Perimeter

Circles, Arcs & Sectors

Volume & Surface Area

Right-Angled Triangles - Pythagoras & Trigonometry

Sine, Cosine Rule & Area of Triangles

3D Pythagoras & Trigonometry

11. Probability

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Simple Probability Diagrams

Tree Diagrams

Combined & Conditional Probability

12. Statistics

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Statistical Diagrams

Histograms

Cumulative Frequency & Box Plots

Scatter Graphs & Correlation