We can start by using the identity:
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
Expanding the left-hand side of the equation gives:
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca + 2ab + 2bc - 2ca
Simplifying, we get:
(a + b + c)^2 = a^2 + b^2 + c^2 + 4ab + 4bc
We can now substitute the given values of a^2 + b^2 + c^2 and a + b - c into this equation to solve for ab + bc:
(a + b + c)^2 = a^2 + b^2 + c^2 + 4ab + 4bc
(5 + c)^2 = 29 + 4(ab + bc)
25 + 10c + c^2 = 29 + 4(ab + bc)
c^2 + 10c - 4(ab + bc) + 4 = 0
We can now use the quadratic formula to solve for c:
c = (-10 ± sqrt(10^2 - 4*(-4)(a^2 + b^2 + 2ab - 29))) / (2(-4))
Simplifying this expression gives:
c = (-5 ± sqrt(13)) / 2
We can now use the given equation a + b - c = 5 to solve for a + b:
a + b = c + 5
Substituting the two possible values of c, we get:
a + b = (-5 + sqrt(13)) / 2 + 5 = (5 + sqrt(13)) / 2
or
a + b = (-5 - sqrt(13)) / 2 + 5 = (5 - sqrt(13)) / 2
We can now use the identity ab - bc - ca = (a + b - c)(b + c - a)(c + a - b) to calculate the desired value:
ab - bc - ca = ((5 + sqrt(13)) / 2)((5 - sqrt(13)) / 2)(-c) + ((5 - sqrt(13)) / 2)(-c)((5 + sqrt(13)) / 2) + (-c)((5 + sqrt(13)) / 2)((5 - sqrt(13)) / 2)
Simplifying this expression gives:
ab - bc - ca = -13/2
Therefore, the value of ab - bc - ca is -13/2.