$\begin{array}{l}1)\,(x^2 + x)^2 - 14(x^2 + x) + 24\\ = t^2 - 14t + 24 \,\,\,\,\,\,\,\,\,(t = x^2 + x)\\ =t^2 - 2t - 12t + 24\\ =t(t - 2) - 12(t - 2)\\ =(t-2)(t-12)\\ =(x^2 + x -2)(x^2 + x - 12)\\ 2) \, x^4 + 2x^3 + 5x^2 + 4x - 12\\ = x^4 - x^3 + 3x^3 - 3x^2 + 8x^2 - 8x + 12x -12\\ = x^3(x - 1) + 3x^2(x - 1) + 8x(x - 1) + 12(x - 1)\\ = (x-1)(x^3 + 3x^2 + 8x + 12)\\ 3)\,(x+1)(x+3)(x+5)(x+7) + 15\\ =[(x+1)(x+7)][(x+3)(x+5)] + 15\\ = (x^2 + 8x + 7)(x^2 + 8x + 15) + 15\\ = (t + 7)(t + 15) + 15\,\,\,\,\,\,\,\,\,\,\,\,(t = x^2 + 8x)\\ = t^2 + 22t + 105 + 15\\ = t^2 + 10t + 12t + 120\\ =t(t + 10) + 12(t + 10)\\ =(t + 10)(t + 12)\\ = (x^2 + 8x + 10)(x^2 + 8x + 12)\\ 4) \, (x^2 + x)^2 + 4x^2 + 4x -12\\ = (x^2 + x)^2 + 4(x^2 + x) - 12\\ = t^2 + 4t - 12\,\,\,\,\,\,\,\,\,\,\,\,(t = x^2 + x)\\ = t^2 - 2t + 6t -12\\ = t(t -2) +6(t - 2)\\ = (t - 2)(t + 6)\\ = (x^2 + x -2)(x^2 + x + 6)\\ 5) \, (x+1)(x+2)(x+3)(x+4) + 1\\ = [(x+1)(x+4)][(x+2)(x+3)] +1\\ = (x^2 + 5x + 4)(x^2 + 5x + 6) + 1\\ = (t + 4)(t+ 6) + 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,(t = x^2 + 5x)\\ = t^2 + 10t + 25\\ = (t + 5)^2\\=(x^2 + 5x + 5)^2\end{array}$
