Let’s resolve into factors

(x + 1) (x + 9) (x + 5)² + 63

= (x² + 9x + x + 9) (x + 5)² + 63

Apply the formula (a + b)² = a² + b² + 2ab in (x + 5)²

= (x² + 10x + 9) (x² + 10x + 25) + 63

Let x² + 10x = a

= (a + 9)(a + 25) + 63

= a² + 25a + 9a + 225 + 63

= a² + 34a + 288

= a² + 18a + 16a + 288

= a(a + 18) + 16(a + 18)

= (a + 16)(a + 18)

Putting back the values of a we get,

= (x² + 10x + 16)( x² + 10x + 18)

= (x² + 8x + 2x + 16)(x² + 10x + 18)

= [x(x + 8) + 2(x + 8)] (x² + 10x + 18)

= (x + 8)(x + 2) (x² + 10x + 18)