answer:**Step 1:** Start by completing the square for x. The expression can be rewritten as: [ x^2 + 4xy + 5y^2 - 8x - 6y = (x + 2y)^2 - 4y^2 + 5y^2 - 8x - 6y. ] This simplifies to: [ (x + 2y)^2 + y^2 - 8x - 6y. ] **Step 2:** Minimize the expression in terms of x, set the derivative with respect to x of (x + 2y - 4)^2 to zero: [ x + 2y - 4 = 0 implies x = 4 - 2y. ] **Step 3:** Substitute x = 4 - 2y back into the expression: [ (4 - 2y + 2y)^2 + y^2 - 8(4 - 2y) - 6y = 16 + y^2 - 32 + 16y - 6y = y^2 + 10y - 16. ] Complete the square for y: [ y^2 + 10y - 16 = (y + 5)^2 - 25 - 16 = (y + 5)^2 - 41. ] The minimum value is boxed{-41}, occurring when y = -5, and thus x = 4 - 2(-5) = 14.

answer:First, we convert the numbers from base-3 and base-12 to base-10. For 5triangle_3: [ 5triangle_3 = 5cdot3^1 + trianglecdot3^0 = 15 + triangle ] For triangle4_{12}: [ triangle4_{12} = trianglecdot12^1 + 4cdot12^0 = 12triangle + 4 ] Equating and solving for triangle: [ 15 + triangle = 12triangle + 4 ] [ 15 - 4 = 12triangle - triangle ] [ 11 = 11triangle ] [ triangle = frac{11}{11} ] [ boxed{1} = triangle ] Conclusion: By converting each number to base-10 and setting up an equation, we found that triangle = 1.

answer:Since point M is the midpoint of line segment AB, the length of AM is half the length of AB. Therefore, the length of AM is 12 cm / 2 = 6 cm. Now, since point N is the midpoint of line segment AM, the length of NM is half the length of AM. Therefore, the length of NM is 6 cm / 2 = 3 cm. So, the length of segment NM is boxed{3} cm.

answer:Mr. Salazar initially had a total of 7 times 12 = 84 oranges. From these, he reserved a quarter for his friend, which amounts to 84 times frac{1}{4} = 21 oranges. This leaves him with 84 - 21 = 63 oranges intended to be sold yesterday. Out of these, he managed to sell 63 times frac{3}{7} = 27 oranges yesterday. Therefore, before considering the rotten oranges, he had 63 - 27 = 36 oranges left. After discovering four rotten oranges, the total number of oranges left to be sold today is 36 - 4 = 32. Thus, the final answer is boxed{32}.