Factorise the following:

x⁴ + 5x² + 9

x⁴ + 5x² + 9 = x⁴ + 6x² - x² + 9

= (x⁴ + 6x² + 9) - x²

= [(x²)² + 2(3x²) + 3²] - x²

We know that,

(a + b)² = a² + 2ab + b²

and

a² - b² = (a - b)(a + b)

∴ [(x²)² + 2(3x²) + 3²] - x² = (x² + 3)² - x²

= (x² + 3 - x)(x² + 3 + x)

Hence, x⁴ + 5x² + 9 = (x² - x + 3)(x² + x + 3)