Easy way to see that $(x^2 + 5x + 4)(x^2 + 5x + 6) - 48 = (x^2 + 5x + 12)(x^2 + 5x - 2)$?
$$(x^2 + 5x + 5-1)(x^2 + 5x + 5+1) - 48 = (x^2 + 5x + 5)^2-7^2$$
For $a=x^2+5x+4$ we get to factor $a(a+2)-48=a^2+2a-48=(a+8)(a-6)$.
$$(x^2 + 5x + 5-1)(x^2 + 5x + 5+1) - 48 = (x^2 + 5x + 5)^2-7^2$$
For $a=x^2+5x+4$ we get to factor $a(a+2)-48=a^2+2a-48=(a+8)(a-6)$.