<em>R(x)</em> is a polynomial of degree 7, so it has up to 7 distinct complex roots <em>r</em>₁, ..., <em>r</em>₇, and we can write it in terms of these roots as
<em>R(x)</em> = (<em>x</em> - <em>r</em>₁) (<em>x</em> - <em>r</em>₂) ... (<em>x</em> - <em>r</em>₇)
The coefficients of <em>R(x)</em> are all real, so the roots must all be complex numbers, and any of these roots with non-zero imaginary parts must occur along with their complex conjugates. This means if <em>a</em> + <em>b</em> <em>i</em> is a root, then is <em>a</em> - <em>b</em> <em>i</em> is also a root.
(a) We're told that -5 - 3<em>i</em> and 2<em>i</em> are roots to <em>R(x)</em>, so we also know that -5 + 3<em>i</em> and -2<em>i</em> are roots.
There are 4 roots accounted for, leaving us with 3 unknown roots. These roots cannot all be non-real, because we can only count 2 of them as a conjugate pair. So we can have either
(b) at most 3 real roots, or
(c) at most 2 non-real roots and 1 real root.