Advent Challenge 3: Time to remanufacture the presents!
JavaScript (ES6), 58 bytes

a=&gt;a.reduce((p,[c,x,y,z])=&gt;p+c+6*(y*z+x*(y+=z))+4*(x+y),0)


Test cases



let f =

a=&gt;a.reduce((p,[c,x,y,z])=&gt;p+c+6*(y*z+x*(y+=z))+4*(x+y),0)

console.log(f([[7, 8, 6, 7], [7, 7, 5, 5], [8, 9, 6, 7], [6, 5, 10, 10], [5, 9, 6, 7], [9, 9, 10, 6], [8, 10, 10, 6], [6, 5, 7, 9], [7, 10, 8, 8], [5, 9, 9, 10]])) // 11866
console.log(f([[5, 10, 8, 9], [8, 8, 5, 8], [8, 7, 7, 6], [5, 9, 9, 10], [9, 7, 5, 8], [9, 8, 9, 5], [7, 5, 6, 7], [5, 7, 6, 10]])) // 8854
console.log(f([[9, 8, 8, 8], [10, 9, 8, 5], [10, 7, 5, 5], [10, 10, 6, 6], [8, 5, 8, 7]])) // 4853
console.log(f([[7, 7, 8, 10], [8, 10, 7, 8], [9, 7, 7, 8], [8, 5, 10, 5], [6, 6, 6, 8], [8, 9, 7, 5], [8, 5, 6, 5], [7, 9, 8, 5], [10, 10, 10, 8]])) // 9717
console.log(f([[5, 8, 9, 7], [5, 8, 7, 10], [5, 7, 7, 6], [5, 5, 5, 6], [9, 9, 5, 7], [5, 6, 7, 8], [8, 5, 8, 7], [6, 9, 5, 5], [10, 10, 9, 10]])) // 9418
console.log(f([[9, 9, 7, 10], [5, 8, 7, 9], [5, 5, 9, 8], [10, 5, 9, 10], [8, 5, 10, 7], [8, 9, 5, 5], [5, 10, 6, 10]])) // 8178
console.log(f([[5, 9, 5, 8], [7, 8, 10, 6], [7, 10, 7, 10], [8, 9, 7, 5], [5, 7, 8, 6], [9, 9, 6, 10], [6, 5, 9, 9], [7, 9, 9, 9]])) // 9766
console.log(f([[7, 10, 5, 10], [8, 10, 8, 9], [8, 6, 7, 8], [6, 9, 8, 5], [6, 7, 10, 9], [7, 6, 5, 8]])) // 7118
console.log(f([[10, 6, 7, 5], [5, 9, 5, 9], [9, 7, 8, 5], [6, 6, 9, 9], [9, 9, 6, 9], [10, 5, 8, 9], [7, 5, 6, 10], [9, 10, 5, 5]])) // 8007
console.log(f([[8, 10, 7, 8], [9, 10, 5, 8], [6, 7, 5, 6], [10, 10, 9, 8], [7, 5, 8, 9], [10, 10, 6, 7], [10, 8, 9, 10], [5, 10, 5, 5]])) // 9331




How?

The only trick used here is to factorize (xy + xz) as x(y + z) and re-use the sum (y + z) in the last part of the formula.

a =&gt; a.reduce(                    // for each present in a:
  (s, [c, x, y, z]) =&gt;            //   s = sum, [c, x, y, z] = present parameters
    s +                           //   add to s:
    c +                           //     c
    6 * (y * z + x * (y += z)) +  //     6(yz + x(y + z))
    4 * (x + y),                  //     4(x + (y + z))
  0                               //   initial sum = 0
)                                 // end of reduce()