a⁴ + a²b² + b⁴  =  (a² + ab + b²)(a² − ab + b²)
a⁴ + a²b² + b⁴  =  (a² + b²)² − (ab)²

📐 a⁴ + a²b² + b⁴ — Derivation

a⁴ + a²b² + b⁴ = (a² + ab + b²)(a² − ab + b²)

Identities Used

Perfect Square
(a² + b²)² = a⁴ + 2a²b² + b⁴
∴ (a+b)² − a² + 2ab + b²
Difference of Squares
a² − b² = (a + b)(a − b)
Here X = a²+b², Y = ab

Step-by-Step Derivation

1
a⁴ + a²b² + b⁴
Original expression
2
= (a²)² + 2a²b² − a²b² + (b²)²
Write a²b² = 2a²b² − a²b² to build a perfect square
3
= (a² + b²)² − (ab)²
Since (a²)²+2a²b²+(b²)² = (a²+b²)²   and   a²b² = (ab)²
4
= (a² + b² + ab)(a² + b² − ab)
Difference of squares: X²−Y² = (X+Y)(X−Y), X=a²+b², Y=ab
5
= (a² + ab + b²)(a² − ab + b²)
Rearrange in standard descending order
∴   a⁴ + a²b² + b⁴ = (a² + ab + b²)(a² − ab + b²)

🧮 Factorization Practice

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