x² + 8x + 16 — Use (a+b)² = a² + 2ab + b² with a = x and b = 4: x² + 8x + 16.

x² − 10x + 25 — Use (a−b)² = a² − 2ab + b² with a = x and b = 5: x² − 10x + 25.

(x+5)(x−5) — A difference of squares: a² − b² = (a+b)(a−b). Here 25 = 5², so the answer is (x+5)(x−5).

Expansion & Factoring

Q: Expand (x + 3)(x − 3).

x² − 9 — This is the difference of squares: (a+b)(a−b) = a² − b². With a = x and b = 3 you get x² − 9.

Q: Factor x² + 7x + 12.

(x+3)(x+4) — You need a pair that multiplies to 12 and adds to 7. Since 3 × 4 = 12 and 3 + 4 = 7, the answer is (x+3)(x+4).

Q: Factor x² − 9x + 20.

(x−4)(x−5) — The pair must multiply to 20 and add to −9, so both are negative: −4 and −5. That gives (x−4)(x−5).

Q: Take out the greatest common factor: 6x² + 9x = ?

3x(2x + 3) — Both terms share 3x, and that is the largest common factor, so pull it out: 3x(2x + 3).

Q: Expand (2x + 1)(x + 3).

2x² + 7x + 3 — Multiply every pair of terms: 2x·x + 2x·3 + 1·x + 1·3 = 2x² + 6x + x + 3 = 2x² + 7x + 3.

Where this is used

Factoring is more than a trick for tidying up homework. RSA encryption keeps your messages safe purely because nobody knows a fast way to split a huge number back into its prime factors. Optimizing a machine learning model means rewriting its loss as a product so the minimum falls out. And the expansion rule (a+b)² = a² + 2ab + b² grows up into the binomial theorem, which sits under most of probability and physics.

Key points

The Three Expansion Formulas

Three patterns are worth knowing by heart: (1) (a+b)² = a² + 2ab + b², (2) (a−b)² = a² − 2ab + b², (3) (a+b)(a−b) = a² − b². Notice the middle term disappears in the third one — that is what makes it so useful.

Expanding (x+3)²

Take the first formula with a = x and b = 3, and substitute straight in.

What Factoring Means

Factoring means rewriting a polynomial as a product. Three moves cover most cases: pull out a common factor, run one of the expansion formulas backwards, or hunt for a pair of numbers that fits.

Finding the Number Pair

For x² + bx + c, look for two numbers p and q that multiply to c and add to b. Then the answer is simply (x+p)(x+q).

Using Factoring to Solve Equations

Factor x² + 5x + 6 = 0 into (x+2)(x+3) = 0. If a product is zero, at least one of the parts must be zero, so x = −2 or x = −3.

Check yourself 10questions

Tap an option to check

Q1 Expand (x + 4)².

Q2 Expand (x − 5)².

Q3 Expand (x + 3)(x − 3).

Q4 Factor x² + 7x + 12.

Q5 Factor x² − 9x + 20.

Q6 Take out the greatest common factor: 6x² + 9x = ?

Q7 Factor x² − 25.

Q8 Expand (2x + 1)(x + 3).

Q9 Factor x² + 2x − 15.

Q10 Factor 4x² − 4x + 1.

Jobs that use this

Cryptography Engineer / Security Researcher $150k

Machine Learning Engineer $150k

Quantum Computing Researcher $180k

Cryptography Engineer / Security Researcher$150k

Designs secure communication systems such as RSA and elliptic-curve cryptography, built on the theory of prime factorization and polynomials.

Maths used: FactoringNumber theoryGroup theoryProbability

Machine Learning Engineer$150k

Optimizes loss functions and rearranges regularization terms — expanding and factoring expressions is what makes the numerical work run faster.

Maths used: FactoringDifferentiationLinear algebraOptimization theory

Quantum Computing Researcher$180k

Researches and implements the theory behind Shor's algorithm, which lets a quantum computer factor large numbers.

Maths used: FactoringNumber theoryQuantum mechanicsLinear algebra