If $\frac{13x+43}{2x^2+17x+30}=\frac{A}{2x+5}+\frac{B}{x+6}$, then $A^2+B^2=$

Given:

$\frac{13x+43}{2x^2+17x+30}=\frac{A}{2x+5}+\frac{B}{x+6}$

$\Rightarrow \frac{13x+43}{2x^2+17x+30}=\frac{A\left(x+6\right)+B\left(2x+5\right)}{\left(2x+5\right)\left(x+6\right)}$

$\Rightarrow \frac{13x+43}{2x^2+17x+30}=\frac{\left(A+2B\right)x+6A+5B}{2x^2+17x+30}$

$\Rightarrow \left(A+2B\right)x+6A+5B=13x+43$

On comparing, we get

$A+2B=13 ....\left(\mathrm{i}\right)$

$6A+5B=43 ....\left(\mathrm{ii}\right)$

On solving $\left(\mathrm{i}\right) \& \left(\mathrm{ii}\right)$, we get

$7B=35\Rightarrow B=5$

$A=13-2B=3$

So,

$A^2+B^2=9+25=34$