1. Stating the problem: Simplify the expressions $A = \sqrt{12} - 4\sqrt{27}$ and $B = 2\sqrt{50} + 4\sqrt{32}$.\n\n2. Formula and rules: Recall that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ and simplify square roots by factoring out perfect squares.\n\n3. Simplify $A$: \n$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$\n$4\sqrt{27} = 4 \times \sqrt{9 \times 3} = 4 \times 3\sqrt{3} = 12\sqrt{3}$\nSo, $A = 2\sqrt{3} - 12\sqrt{3} = (2 - 12)\sqrt{3} = -10\sqrt{3}$.\n\n4. Simplify $B$: \n$2\sqrt{50} = 2 \times \sqrt{25 \times 2} = 2 \times 5\sqrt{2} = 10\sqrt{2}$\n$4\sqrt{32} = 4 \times \sqrt{16 \times 2} = 4 \times 4\sqrt{2} = 16\sqrt{2}$\nSo, $B = 10\sqrt{2} + 16\sqrt{2} = (10 + 16)\sqrt{2} = 26\sqrt{2}$.\n\nFinal answers: \n$A = -10\sqrt{3}$\n$B = 26\sqrt{2}$