jmc

We can factor $x^2+3x+2=(x+1)(x+2)$ and $x^2+7x+12=(x+3)(x+4).$ Then the given polynomial is $$\begin{array}{rl}(x+1)(x+2)(x+3)(x+4)+(x^2+5x-6)& =(x+1)(x+4)(x+2)(x+3)+(x^2+5x-6)\\ & =(x^2+5x+4)(x^2+5x+6)+(x^2+5x-6).\end{array}$$Let $y=x^2+5x.$ Then $$\begin{array}{rl}(x^2+5x+4)(x^2+5x+6)+(x^2+5x-6)& =(y+4)(y+6)+(y-6)\\ & =y^2+10y+24+y-6\\ & =y^2+11y+18\\ & =(y+2)(y+9)\\ & =\boxed{(x^2+5x+2)(x^2+5x+9)}.\end{array}$$