Going to be beautiful

Algebra Level 5

The equation (z+1)7+z7=0{ (z+1) }^{ 7 }+{ z }^{ 7 }=0 has roots z1,z2z7.{ z }_{ 1 },{ z }_{ 2 }\cdots{ z }_{ 7 }. Let r=17Re(zr)=a \sum _{ r=1 }^{ 7 }{ \text{Re}({ z }_{ r }) } =a and r=17Im(zr)=b. \sum _{ r=1 }^{ 7 }{ \text{Im}({ z }_{ r }) } =b. Then the quadratic equation whose roots are 2a\left| 2a \right| and b+9\left| b+9 \right| can be represented as x2+cx+d=0.{ x }^{ 2 }+cx+d=0. Find the value of (c+d+21).(\left| c \right| +\left| d \right| +21).

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