Oxford University MAT 2015 Solved Papers

H. Recall that ${\log }_{\mathit{a}}\left(\mathit{b}\right)=\mathit{c}$ describes the exponential relationship between $\mathit{a},\mathit{b}$, and $\mathit{c}$ - namely that ${\mathit{a}}^{\mathit{c}}=\mathit{b}$. Hence, the equation becomes
$4-5{\mathit{x}}^2-6{\mathit{x}}^3={\left({\mathit{x}}^2+2\right)}^2.$
Rearranging and cancelling terms gives
${\mathit{x}}^4+6{\mathit{x}}^3+9{\mathit{x}}^2=0.$
This factorises to
${\mathit{x}}^2{\left(\mathit{x}+3\right)}^2=0.$
This has two roots and so the answer is (c).