Expansion Intuition Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Expansion Intuition.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Understanding algebraic expansion as the process of applying the distributive property to multiply out factors and remove parentheses.

Open up the parentheses: (x+2)(x+3)(x + 2)(x + 3) becomes x2+3x+2x+6=x2+5x+6x^2 + 3x + 2x + 6 = x^2 + 5x + 6.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Expansion intuition distributes products across sums until no parentheses remain.

Common stuck point: The procedure for expansion intuition is the easy part; the trap is dropping the cross term when squaring. Asking "Am I turning a product of factors into a sum of terms by distributing?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I turning a product of factors into a sum of terms by distributing?

Worked Examples

Example 1

easy
Expand (x+3)(x+5)(x + 3)(x + 5).

Answer

x2+8x+15x^2 + 8x + 15

First step

1
Use FOIL: First: x⋅x=x2x \cdot x = x^2. Outer: x⋅5=5xx \cdot 5 = 5x. Inner: 3⋅x=3x3 \cdot x = 3x. Last: 3⋅5=153 \cdot 5 = 15.

Full solution

  1. 2
    Combine: x2+5x+3x+15=x2+8x+15x^2 + 5x + 3x + 15 = x^2 + 8x + 15.
  2. 3
    Notice: 3+5=83 + 5 = 8 (middle coefficient) and 3×5=153 \times 5 = 15 (constant).
Expansion distributes each term in one factor to every term in the other. FOIL is a mnemonic for the four products when multiplying two binomials.

Example 2

medium
Expand (2x−3)2(2x - 3)^2.

Example 3

medium
Expand (x+2)(x2+3x+1)(x + 2)(x^2 + 3x + 1).

Example 4

medium
Expand (x+1)2+(x−1)2(x + 1)^2 + (x - 1)^2.

Example 5

challenge
Without full expansion, find the coefficient of xx in (x+1)(x+2)(x+3)(x+4)(x + 1)(x + 2)(x + 3)(x + 4).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Expand (x−2)(x+7)(x - 2)(x + 7).

Example 2

medium
Expand (a+b)2(a + b)^2.

Example 3

easy
Expand 2(x+5)2(x+5).

Example 4

easy
Expand (x+2)(x+3)(x+2)(x+3).

Example 5

easy
Expand (x+4)2(x+4)^2.

Example 6

easy
Expand −3(x−2)-3(x-2).

Example 7

easy
Expand x(x+7)x(x+7).

Example 8

easy
Expand (x−5)(x+5)(x-5)(x+5).

Example 9

easy
Expand 5(2x+3)5(2x+3).

Example 10

easy
Expand (x+1)(x−2)(x+1)(x-2).

Example 11

medium
Expand (2x+3)(x−4)(2x+3)(x-4).

Example 12

medium
Expand (x+2)3(x+2)^3.

Example 13

medium
Expand and simplify 2(x+3)+3(x−1)2(x+3)+3(x-1).

Example 14

medium
Expand (x+y)2−(x−y)2(x+y)^2-(x-y)^2.

Example 15

medium
Expand (a+b+c)2(a+b+c)^2.

Example 16

medium
Expand (2x−1)(2x+1)(2x-1)(2x+1).

Example 17

medium
Compute 103×97103\times97 by expansion.

Example 18

challenge
Show (a+b)2=a2+b2(a+b)^2=a^2+b^2 is false in general and find when it holds.

Example 19

challenge
Expand (x+1)(x+2)(x+3)(x+1)(x+2)(x+3).

Example 20

challenge
Find the coefficient of x2x^2 in (x+1)(x+2)(x+3)(x+4)(x+1)(x+2)(x+3)(x+4) without full expansion.

Example 21

medium
Expand (3x−2)(x+5)(3x-2)(x+5).

Example 22

medium
Expand (x−3)2(x-3)^2.

Example 23

easy
Expand 3(x+4)3(x + 4).

Example 24

easy
Expand −4(2x−5)-4(2x - 5).

Example 25

easy
Expand (x−3)(x−4)(x - 3)(x - 4).

Example 26

easy
Expand (x+6)(x−6)(x + 6)(x - 6).

Example 27

easy
Expand 4x(x−3)4x(x - 3).

Example 28

easy
Expand (2x+1)(x+4)(2x + 1)(x + 4).

Example 29

medium
Expand (3x+5)(2x−1)(3x + 5)(2x - 1).

Example 30

medium
Expand (3x−4)2(3x - 4)^2.

Example 31

medium
Expand (x+2)(x−3)+(x+1)(x+4)(x + 2)(x - 3) + (x + 1)(x + 4).

Example 32

medium
Expand 5(x−1)−2(3x+4)5(x - 1) - 2(3x + 4).

Example 33

medium
Expand (x+y)(x−y)(x + y)(x - y).

Example 34

medium
Expand (x−2)(x2+2x+4)(x - 2)(x^2 + 2x + 4).

Example 35

medium
Expand −(x−3)(x+5)-(x - 3)(x + 5).

Example 36

hard
Expand (x+3)(x−3)(x2+9)(x + 3)(x - 3)(x^2 + 9).

Example 37

hard
Expand (x+2)4(x + 2)^4 using the binomial theorem.

Example 38

hard
Use (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2 to compute 52⋅4852 \cdot 48 mentally.

Example 39

hard
Expand (x−y)3(x - y)^3.

Example 40

challenge
Expand (x+y+z)2−(x−y−z)2(x + y + z)^2 - (x - y - z)^2.

Example 41

challenge
Find the constant term in (x+2)(x−1)(x+3)(x + 2)(x - 1)(x + 3).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

distributive property