Solve y=a(x+1)(x+3)

$$y = a ( x + 1 ) ( x + 3 )$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{y}{\left(x+1\right)\left(x+3\right)}\text{, }&x\neq -3\text{ and }x\neq -1\\a\in \mathrm{C}\text{, }&\left(x=-1\text{ or }x=-3\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a by x+1.

$$y=\left(ax+a\right)\left(x+3\right)$$

Use the distributive property to multiply ax+a by x+3 and combine like terms.

$$y=ax^{2}+4ax+3a$$

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

$$ax^{2}+4ax+3a=y$$

Combine all terms containing a.

$$\left(x^{2}+4x+3\right)a=y$$

Divide both sides by x^{2}+4x+3.

$$\frac{\left(x^{2}+4x+3\right)a}{x^{2}+4x+3}=\frac{y}{x^{2}+4x+3}$$

Dividing by x^{2}+4x+3 undoes the multiplication by x^{2}+4x+3.

$$a=\frac{y}{x^{2}+4x+3}$$

Divide y by x^{2}+4x+3.

$$a=\frac{y}{\left(x+1\right)\left(x+3\right)}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{y}{\left(x+1\right)\left(x+3\right)}\text{, }&x\neq -3\text{ and }x\neq -1\\a\in \mathrm{R}\text{, }&\left(x=-1\text{ or }x=-3\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a by x+1.

$$y=\left(ax+a\right)\left(x+3\right)$$

Use the distributive property to multiply ax+a by x+3 and combine like terms.

$$y=ax^{2}+4ax+3a$$

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

$$ax^{2}+4ax+3a=y$$

Combine all terms containing a.

$$\left(x^{2}+4x+3\right)a=y$$

Divide both sides by x^{2}+4x+3.

$$\frac{\left(x^{2}+4x+3\right)a}{x^{2}+4x+3}=\frac{y}{x^{2}+4x+3}$$

Dividing by x^{2}+4x+3 undoes the multiplication by x^{2}+4x+3.

$$a=\frac{y}{x^{2}+4x+3}$$

Divide y by x^{2}+4x+3.

$$a=\frac{y}{\left(x+1\right)\left(x+3\right)}$$

Solve for x (complex solution)

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

Solve for x

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