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On examining coefficients, rational roots are possibly factors of +10 or -10, ie.,+1,+2,+5,+10,-1,-2,-5-10.

Checking with x =1, P(1)  = 0,  Hence (x-1) is a factor of P(x),
Checking with x = 2, P(2)=0, Hence (x-2) is a factor too.

P(x) = (x-1)(x-2)(x^2+ax+5) (x^2-3x+2)(x^2+ax+5)

Coefficient of x^3 will be -3+a = -6  ,  hence a = -3

Factors are : P(x) = (x-1)(x-2)(x^2-3x+5)
the quadratic expression has imaginary roots, as 3^2-4*5 <0.  Hence there are no real factors.

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