Solve =a^4/2+a^3+2b^3-11a-20=P(a)

$$= \frac { a ^ { 4 } } { 2 } + a ^ { 3 } + 2 b ^ { 3 } - 11 a - 20 = P ( a )$$

Solve for P

$$\left\{\begin{matrix}P=\frac{a^{4}+2a^{3}-22a+4b^{3}-40}{2a}\text{, }&a\neq 0\\P\in \mathrm{R}\text{, }&b=2\sqrt[3]{5}\times \frac{4^{\frac{2}{3}}}{4}\text{ and }a=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$a^{4}+2a^{3}+4b^{3}-22a-40=2Pa$$

Swap sides so that all variable terms are on the left hand side.

$$2Pa=a^{4}+2a^{3}+4b^{3}-22a-40$$

The equation is in standard form.

$$2aP=a^{4}+2a^{3}-22a+4b^{3}-40$$

Divide both sides by 2a.

$$\frac{2aP}{2a}=\frac{a^{4}+2a^{3}-22a+4b^{3}-40}{2a}$$

Dividing by 2a undoes the multiplication by 2a.

$$P=\frac{a^{4}+2a^{3}-22a+4b^{3}-40}{2a}$$

Divide a^{4}+2a^{3}+4b^{3}-22a-40 by 2a.

$$P=\frac{a^{3}}{2}+\frac{2b^{3}-20}{a}+a^{2}-11$$