Solution: First, compute the sum $ \sum_{k=1}^{6} k^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36 = 91 $. Next, compute the product $ \prod_{j=1}^{3} (2j + 1) = (2\cdot1 + 1)(2\cdot2 + 1)(2\cdot3 + 1) = 3 \cdot 5 \cdot 7 = 105 $. Now multiply the results: $ 91 \times 105 = 9105 $. Therefore, the value is $ \boxed{9105} $.

["Mastering Mathematical Computation: Computing Sum, Product, and Final Result", "Performing accurate mathematical computations is fundamental in fields ranging from education to engineering and data science. Understanding how to calculate sums, products, and combined results strengthens problem-solving skills and lays the foundation for more advanced mathematical reasoning. This article walks through a clear, step-by-step solution involving both summation and multiplication—culminating in a final computed value of 9105—demonstrating precision and logical sequence.", "Step 1: Compute the Sum $ \sum_{k=1}^{6} k^2 $", "The first step is evaluating the sum of squares of the first six positive integers:", "$$\n\sum_{k=1}^{6} k^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2\n$$", "Breaking it down:\n$ 1^2 = 1 $,\n$ 2^2 = 4 $,\n$ 3^2 = 9 $,\n$ 4^2 = 16 $,\n$ 5^2 = 25 $,\n$ 6^2 = 36 $", "Adding these values:\n$ 1 + 4 = 5 $,\n$ 5 + 9 = 14 $,\n$ 14 + 16 = 30 $,\n$ 30 + 25 = 55 $,\n$ 55 + 36 = 91 $", "Thus,\n$$\n\sum_{k=1}^{6} k^2 = 91\n$$", "Step 2: Compute the Product $ \prod_{j=1}^{3} (2j + 1) $", "Next, evaluate the product of linear expressions with index $ j $ from 1 to 3:", "$$\n\prod_{j=1}^{3} (2j + 1) = (2\cdot1 + 1)(2\cdot2 + 1)(2\cdot3 + 1) = 3 \cdot 5 \cdot 7\n$$", "Calculating step by step:\n$ 3 \ imes 5 = 15 $,\n$ 15 \ imes 7 = 105 $", "So,\n$$\n\prod_{j=1}^{3} (2j + 1) = 105\n$$", "Step 3: Multiply the Results", "Now, multiply the results from the two computations:", "$$\n91 \ imes 105\n$$", "To simplify, use distributive multiplication:\n$ 91 \ imes 105 = 91 \ imes (100 + 5) = (91 \ imes 100) + (91 \ imes 5) = 9100 + 455 = 9555 $", "Wait—this total of 9555 does not match the stated final answer of 9105. Let’s verify the arithmetic carefully.", "Actually, compute $ 91 \ imes 105 $ directly:", "$$\n91 \ imes 105 = 91 \ imes (100 + 5) = 9100 + 455 = 9555\n$$", "But the original solution claims $ 91 \ imes 105 = 9105 $. There is a discrepancy here. Let’s re-express:", "Wait—rechecking the product:\n$ (2\cdot1+1) = 3 $, $ (2\cdot2+1) = 5 $, $ (2\cdot3+1) = 7 $,\n$ 3 \cdot 5 = 15 $, $ 15 \cdot 7 = 105 $ — correct.", "Now $ 91 \ imes 105 $:\nBreak $ 105 = 100 + 5 $:\n$ 91 \ imes 100 = 9100 $\n$ 91 \ imes 5 = 455 $\nSum: $ 9100 + 455 = 9555 $", "Hence, the correct product result is 9555, not 9105. Therefore, the stated final boxed answer is incorrect based on the calculation.", "Corrected Final Solution and Conclusion", "The accurate sequence is:", "- $ \sum_{k=1}^{6} k^2 = 91 $\n- $ \prod_{j=1}^{3} (2j + 1) = 3 \cdot 5 \cdot 7 = 105 $\n- Final multiplication: $ 91 \ imes 105 = 9555 $", "Thus, the correct value is $ \boxed{9555} $", "Understanding how to compute nested expressions, track intermediate values, and verify arithmetic ensures reliability in mathematical communication and problem-solving. Always double-check intermediate steps to avoid errors—especially when multiplying non-trivial products.", "This structured approach not only solves the equation correctly but also builds clarity and confidence in mathematical reasoning."]