Explanation

(c)
$\sin{A + \cos{A = m}}$ [given]
$\Rightarrow \ \sin^{3}{A + \cos^{3}{\ A + 3\cos{A\sin A}}}$
$(\sin{A + \cos A) = m^{3}}$
$\Rightarrow \ n + 3m\sin{A\cos{A = m^{3}}}$ ….(i)
$\lbrack\because\sin^{3}{A + \cos^{3}{A = n}}\rbrack$
Again, $\sin{A + \cos{A = m}}$
$\Rightarrow \ \ \sin^{2}{A + \cos^{2}{A + 2sin\ A\cos{A = m^{2}}}}$
$\Rightarrow \ \sin{A\cos{A =}}\frac{m^{2} - 1}{2}$ …(ii)
From Eqs. (i) and (ii), we get
$n + 3m\frac{(m^{2} - 1)}{2} = m^{3}$
$\Rightarrow \ \ \ 2n + 3m^{3} - 3m = 2m^{3}$
$\Rightarrow \ \ m^{3} - 3m + 2n = 0$