Solve 41(2a+b)(2a+b)+b(2a+b)-4a^2b=b

$$41 ( 2 a + b ) ( 2 a + b ) + b ( 2 a + b ) - 4 a ^ { 2 } b = b$$

Solve for a

$$\left\{\begin{matrix}a=\frac{\sqrt{b}\left(\sqrt{168b^{2}-3b+164}-83\sqrt{b}\right)}{4\left(41-b\right)}\text{; }a=-\frac{\sqrt{b}\left(\sqrt{168b^{2}-3b+164}+83\sqrt{b}\right)}{4\left(41-b\right)}\text{, }&b\neq 41\text{ and }b\geq 0\\a=-\frac{1721}{166}\approx -10.36746988\text{, }&b=41\end{matrix}\right.$$

Solve for b

$$b=\frac{\sqrt{16a^{4}-1328a^{3}+12a^{2}-332a+1}}{84}+\frac{a^{2}}{21}-\frac{83a}{42}+\frac{1}{84}$$

$$b=-\frac{\sqrt{16a^{4}-1328a^{3}+12a^{2}-332a+1}}{84}+\frac{a^{2}}{21}-\frac{83a}{42}+\frac{1}{84}\text{, }16a^{4}-1328a^{3}+12a^{2}-332a+1\geq 0$$