Origin is shifted $(h, k)$
$\therefore$ New coordinate $\left(x^{\prime}, y^{\prime}\right)$
$$
\begin{gathered}
x^{\prime}=x-h, y^{\prime}=y-k \\
\therefore \quad x^2+5 x y+2 y^2+5 x+6 y+7=0 \\
\left(x^{\prime}+h\right)^2+5\left(x^{\prime}+h\right)\left(y^{\prime}+k\right)+2\left(y^{\prime}+k\right)^2 \\
+5\left(x^{\prime}+h\right)+6\left(y^{\prime}+K\right)+7=0 \\
x^{\prime 2}+2 h x^{\prime}+h^2+5 x^{\prime} y^{\prime}+5 k x^{\prime}+5 y^{\prime} h+5 h k+2 y^{\prime 2} \\
+2 k^2+4 y^{\prime} k+5 x^{\prime}+5 h+6 y^{\prime}+6 k+7=0
\end{gathered}
$$
This equation is free from first order
$$
\begin{aligned}
& \therefore & 2 h+5 k+5 & =0 \\
\text { and } & & 5 h+4 k+6 & =0
\end{aligned}
$$
On solving Eqs. (i) and (ii), we get
$$
h=\frac{-10}{17}, k=\frac{-13}{17}
$$