Final answer:
The expression simplifies to a constant value under the given constraints, which after manipulation and substitution of mn+np+pm = 12, results in the value being 3.
Step-by-step explanation:
The question asks for the value of a given algebraic expression under certain constraints. Start by simplifying the expression:
(m(n^2p+p^2)+n(m^2+p^2)+p(m^2+n^2))/(mnp)
We know that m+n+p=8 and mn+np+pm=12. If we look at the numerator m(n^2p) can be combined with n(p^2), and n(m^2) can be combined with p(n^2), which simply turns to mn^2p+mn^2p+mn^2p (since m^2+x^2 unifies to m^2+x^2 when multiplied by the variable not in the square). Each of these terms has mn^2p which is the product of all variables. Hence, we can refactor the numerator to 3mn^2p+p^3+m^3.
If we divide this by mnp, we then simplify this term into 3n+p^2/m^2. Now, since we have mn+np+pm = 12, we can substitute the values in the simplified expression, which results in 3n+p(12-mn)/m or 3n+12p/m-np^2/m. Given the constraints, solving this system of equations results in the expression simplifying to a constant. Since the options given are constants, we can deduce the solution without the need for complete simplification. Doing so, we find that the expression valued at 12/m+12p/m which leads to 24/m if p=1.
Thus, we can conclude from the given options that the correct answer is Option B) 3.