Method 1
Expand the brackets (by multiplying all terms in the first bracket by all terms in the second)

$\sqrt{a} \times \sqrt{a} + \sqrt{a} \times (- 2 \sqrt{b}) + \sqrt{4 b} \times \sqrt{a} + \sqrt{4 b} \times (- 2 \sqrt{b}) = \sqrt{a} \sqrt{a} - 2 \sqrt{a} \sqrt{b} + \sqrt{4 b} \sqrt{a} - 2 \sqrt{4 b} \sqrt{b}$

Simplify $\sqrt{4 b}$ using the surd rule that $\sqrt{x y} = \sqrt{x} \sqrt{y}$

$\sqrt{4 b} = \sqrt{4} \sqrt{b} = 2 \sqrt{b}$

[1]

Replace $\sqrt{4 b}$ with $2 \sqrt{b}$ in the expansion above and cancel the middle terms

$\sqrt{a} \sqrt{a} - 2 \sqrt{a} \sqrt{b} + 2 \sqrt{b} \sqrt{a} - 4 \sqrt{b} \sqrt{b} = \sqrt{a} \sqrt{a} - 4 \sqrt{b} \sqrt{b}$

Simplify $\sqrt{a} \sqrt{a}$ using the surd rule that $\sqrt{x} \sqrt{x} = {(\sqrt{x})}^{2} = x$

$\sqrt{a} \sqrt{a} = a$

[1]

Replace $\sqrt{a} \sqrt{a}$ with $a$ and $\sqrt{b} \sqrt{b}$ with $b$ in the expansion

$a - 4 b$ [1]

Method 2

Simplify $\sqrt{4 b}$ using the surd rule that $\sqrt{x y} = \sqrt{x} \sqrt{y}$

$\sqrt{4 b} = \sqrt{4} \sqrt{b} = 2 \sqrt{b}$

[1]

Replace $\sqrt{4 b}$ with $2 \sqrt{b}$ in the question

$(\sqrt{a} + 2 \sqrt{b}) (\sqrt{a} - 2 \sqrt{b})$

This is the "difference of two squares", (p + q)(p - q) = p² - q²
Use the "difference of two squares" to expand the brackets

${(\sqrt{a})}^{2} - {(2 \sqrt{b})}^{2}$

Square the terms

$a - 2^{2} b$

[1]

$a - 4 b$ [1]

Expanding Brackets (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

Algebra Toolkit

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

Geometry Toolkit

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

Probability Toolkit

Simple Probability Diagrams

Tree Diagrams

Combined & Conditional Probability

12. Statistics

Statistics Toolkit

Statistical Diagrams

Histograms

Cumulative Frequency & Box Plots

Scatter Graphs & Correlation