1. Problem: Calculate the value of the expression $\sqrt{32} + \sqrt{21} - \sqrt{23} + \sqrt{4}$.\n\n2. Formula and rules: We simplify square roots by factoring out perfect squares. For example, $\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$ and $\sqrt{4} = 2$.\n\n3. Simplify each term:\n- $\sqrt{32} = 4\sqrt{2}$\n- $\sqrt{21}$ cannot be simplified further\n- $\sqrt{23}$ cannot be simplified further\n- $\sqrt{4} = 2$\n\n4. Substitute back into the expression:\n$$4\sqrt{2} + \sqrt{21} - \sqrt{23} + 2$$\n\n5. Since $\sqrt{21}$ and $\sqrt{23}$ are irrational and do not simplify to the same terms, the expression cannot be simplified further by combining like terms.\n\n6. Approximate values:\n- $4\sqrt{2} \approx 4 \times 1.414 = 5.656$\n- $\sqrt{21} \approx 4.583$\n- $\sqrt{23} \approx 4.796$\n- $2$ is exact\n\n7. Calculate approximate total:\n$$5.656 + 4.583 - 4.796 + 2 = (5.656 + 4.583) - 4.796 + 2 = 10.239 - 4.796 + 2 = 5.443 + 2 = 7.443$$\n\n8. The closest integer value from the options is 8, but the exact expression is approximately 7.443, so none of the options exactly match. However, since the problem likely expects a simplified exact value, the answer is not a simple integer or $\sqrt{6}$.\n\nTherefore, the expression's value is approximately 7.443, which is closest to option C) 8.