Factorise 3x⁴y³ – 48y³

First take out what both terms share. 3x⁴y³ and 48y³ both contain 3 and y³, since 48 = 3 × 16. So 3x⁴y³ − 48y³ = 3y³(x⁴ − 16).

Now look at x⁴ − 16. This is a difference of two squares, because x⁴ = (x²)² and 16 = 4². Using a² − b² = (a − b)(a + b) with a = x² and b = 4, we get x⁴ − 16 = (x² − 4)(x² + 4).

Do not stop there: x² − 4 is itself a difference of two squares, since 4 = 2². So x² − 4 = (x − 2)(x + 2). The part x² + 4 is a SUM of squares and cannot be factorised any further with real numbers, so it stays as it is.

Putting everything together: 3x⁴y³ − 48y³ = 3y³(x² + 4)(x + 2)(x − 2). You can check it quickly: (x + 2)(x − 2) = x² − 4, and (x² − 4)(x² + 4) = x⁴ − 16, and 3y³ × (x⁴ − 16) = 3x⁴y³ − 48y³.