(b) \( (3+\sqrt{5}) \cdot(4+\sqrt{5}) \)

To tackle the expression \( (3+\sqrt{5}) \cdot(4+\sqrt{5}) \), you can use the distributive property. Multiply each term in the first set of parentheses by each term in the second. So, you would calculate \( 3 \cdot 4 \), \( 3 \cdot \sqrt{5} \), \( \sqrt{5} \cdot 4 \), and \( \sqrt{5} \cdot \sqrt{5} \). Combining all of these gives \( 12 + 3\sqrt{5} + 4\sqrt{5} + 5 \), which simplifies to \( 17 + 7\sqrt{5} \). Testing out parenthetical equations is not only a big part of algebra but also a small peek into how mathematicians develop their ideas. This process can lead to the discovery of new formulas and mathematical principles. So when you multiply those numbers in this way, you're essentially part of a long tradition of mathematicians building upon one another’s work! If you’re ever unsure about your calculations, try using a calculator for verification or breaking down the problem further. Also, common mistakes include miscalculating the square root or forgetting to combine like terms at the end. Always double-check your final expression to avoid those tiny error gremlins!