Since there is one real root and one complex root, there must be one additional real root and another complex root that is the conjugate of the one given.
The conjugate complex roots give rise to the factor
.. (x -2 -5i)*(x -2 +5i)
.. = (x -2)^2 +25
.. = (x^2 -4x +29)
The given real root gives rise to the factor
.. (x +2)
The remaining factor can be written as
.. (ax -3)
since we know the product of the constant terms in these factors must be -174.
The product of these factors is
.. (x +2)(x^2 -4x +29)(ax -3)
.. = ax^4 -(2a +3)x^3 +(21a +6)x^2 +(58a -63)x -174
Matching x-coefficients, we have
.. 53 = 58a -63
.. 116 = 58a
.. 2 = a
(a) The factored function is
.. f(x) = (x +2)(x^2 -4x +29)(2x -3)
(b) The values of a, b, c are ...
.. a = 2
.. b = -(2a +3) = -7
.. c = 21a +6 = 48
(a, b, c) = (2, -7, 48)