Let's create an identity

Algebra Level 5

w=a4+b4+c4+d4x=a2+b2+c2+d2y=ab+ac+ad+bc+bd+cdz=abcd \begin{aligned} w &=& { a }^{ 4 }+{ b }^{ 4 }+{ c }^{ 4 }+{ d }^{ 4 } \\ x &=& { a }^{ 2 }+b^{ 2 }+{ c }^{ 2 }+{ d }^{ 2 } \\ y &=& ab+ac+ad+bc+bd+cd \\ z &=& abcd \\ \end{aligned}

Let a,b,ca,b,c and dd be real numbers; and w,x,yw,x,y and zz some unknown constants such that they satisfy the system of equations above. State the following expression

(a+b+c+d)(a3+b3+c3+d3+abc+abd+bcd+acd)(a+b+c+d)({ a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }+{ d }^{ 3 }+abc+abd+bcd+acd)

in terms of w,x,yw,x,y and zz.

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