Solve ax^2=y(y+2x)^2

$$a x ^ { 2 } = y ( y + 2 x ) ^ { 2 }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\left(\frac{y}{x}+2\right)^{2}y\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+2x\right)^{2}.

$$ax^{2}=y\left(y^{2}+4yx+4x^{2}\right)$$

Use the distributive property to multiply y by y^{2}+4yx+4x^{2}.

$$ax^{2}=y^{3}+4xy^{2}+4yx^{2}$$

The equation is in standard form.

$$x^{2}a=4xy^{2}+y^{3}+4yx^{2}$$

Divide both sides by x^{2}.

$$\frac{x^{2}a}{x^{2}}=\frac{y\left(2x+y\right)^{2}}{x^{2}}$$

Dividing by x^{2} undoes the multiplication by x^{2}.

$$a=\frac{y\left(2x+y\right)^{2}}{x^{2}}$$

Solve for a

$$\left\{\begin{matrix}a=\left(\frac{y}{x}+2\right)^{2}y\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+2x\right)^{2}.

$$ax^{2}=y\left(y^{2}+4yx+4x^{2}\right)$$

Use the distributive property to multiply y by y^{2}+4yx+4x^{2}.

$$ax^{2}=y^{3}+4xy^{2}+4yx^{2}$$

The equation is in standard form.

$$x^{2}a=4xy^{2}+y^{3}+4yx^{2}$$

Divide both sides by x^{2}.

$$\frac{x^{2}a}{x^{2}}=\frac{y\left(2x+y\right)^{2}}{x^{2}}$$

Dividing by x^{2} undoes the multiplication by x^{2}.

$$a=\frac{y\left(2x+y\right)^{2}}{x^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{ay^{3}}+2y^{2}}{a-4y}\text{; }x=\frac{-\sqrt{ay^{3}}+2y^{2}}{a-4y}\text{, }&a\neq 4y\\x=-\frac{y}{4}\text{, }&a=4y\text{ and }y\neq 0\\x\in \mathrm{C}\text{, }&a=0\text{ and }y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{ay^{3}}+2y^{2}}{a-4y}\text{; }x=\frac{-\sqrt{ay^{3}}+2y^{2}}{a-4y}\text{, }&a\neq 4y\text{ and }\left(a\leq 0\text{ or }y\geq 0\right)\text{ and }\left(y\leq 0\text{ or }a\geq 0\right)\\x=-\frac{y}{4}\text{, }&a=4y\text{ and }y\neq 0\\x\in \mathrm{R}\text{, }&a=0\text{ and }y=0\end{matrix}\right.$$