If ax³ + bx² + cx + d is divisible by ax² + c then

${\mathrm{ax}}^3+{\mathrm{bx}}^2+\mathrm{cx}+\mathrm{d}=\left({\mathrm{ax}}^2+\mathrm{c}\right)(\mathrm{\alpha x}+\mathrm{\beta })$ for some $\mathrm{\alpha }, \mathrm{\beta }$

By comparing the constant terms,

we get that $\mathrm{\beta }=\frac{\mathrm{d}}{\mathrm{c}}$

By comparing the coefficients of x²,

we have that $\mathrm{b}=\frac{\mathrm{ad}}{\mathrm{c}}\therefore \mathrm{ad}=\mathrm{bc}$