Solve f(x)=[x^2+(a-1)x+1]e^x*g(x)=-x^2

$$f ( x ) = [ x ^ { 2 } + ( a - 1 ) x + 1 ] e ^ { x } \cdot g ( x ) = - x ^ { 2 }$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{gx^{2}e^{x}-gxe^{x}+x+ge^{x}}{gxe^{x}}\text{, }&g\neq 0\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(x^{2}+ax-x+1\right)e^{x}gx=-x^{2}$$

Use the distributive property to multiply x^{2}+ax-x+1 by e^{x}.

$$\left(x^{2}e^{x}+axe^{x}-xe^{x}+e^{x}\right)gx=-x^{2}$$

Use the distributive property to multiply x^{2}e^{x}+axe^{x}-xe^{x}+e^{x} by g.

$$\left(x^{2}e^{x}g+axe^{x}g-xe^{x}g+e^{x}g\right)x=-x^{2}$$

Use the distributive property to multiply x^{2}e^{x}g+axe^{x}g-xe^{x}g+e^{x}g by x.

$$e^{x}gx^{3}+ae^{x}gx^{2}-e^{x}gx^{2}+e^{x}gx=-x^{2}$$

Subtract e^{x}gx^{3} from both sides.

$$ae^{x}gx^{2}-e^{x}gx^{2}+e^{x}gx=-x^{2}-e^{x}gx^{3}$$

Add e^{x}gx^{2} to both sides.

$$ae^{x}gx^{2}+e^{x}gx=-x^{2}-e^{x}gx^{3}+e^{x}gx^{2}$$

Subtract e^{x}gx from both sides.

$$ae^{x}gx^{2}=-x^{2}-e^{x}gx^{3}+e^{x}gx^{2}-e^{x}gx$$

Reorder the terms.

$$agx^{2}e^{x}=-gx^{3}e^{x}+gx^{2}e^{x}-x^{2}-gxe^{x}$$

The equation is in standard form.

$$gx^{2}e^{x}a=-gx^{3}e^{x}+gx^{2}e^{x}-x^{2}-gxe^{x}$$

Divide both sides by gx^{2}e^{x}.

$$\frac{gx^{2}e^{x}a}{gx^{2}e^{x}}=\frac{x\left(-gx^{2}e^{x}+gxe^{x}-x-ge^{x}\right)}{gx^{2}e^{x}}$$

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

$$a=\frac{x\left(-gx^{2}e^{x}+gxe^{x}-x-ge^{x}\right)}{gx^{2}e^{x}}$$

Divide x\left(-gx^{2}e^{x}+gxe^{x}-x-ge^{x}\right) by gx^{2}e^{x}.

$$a=\frac{-gx^{2}e^{x}+gxe^{x}-x-ge^{x}}{gxe^{x}}$$

Solve for g

$$\left\{\begin{matrix}g=-\frac{x}{\left(x^{2}+ax-x+1\right)e^{x}}\text{, }&\left(x\neq \frac{\sqrt{a^{2}-2a-3}-a+1}{2}\text{ and }x\neq \frac{-\sqrt{a^{2}-2a-3}-a+1}{2}\right)\text{ or }\left(a>-1\text{ and }a<3\right)\\g\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(x^{2}+ax-x+1\right)e^{x}gx=-x^{2}$$

Use the distributive property to multiply x^{2}+ax-x+1 by e^{x}.

$$\left(x^{2}e^{x}+axe^{x}-xe^{x}+e^{x}\right)gx=-x^{2}$$

Use the distributive property to multiply x^{2}e^{x}+axe^{x}-xe^{x}+e^{x} by g.

$$\left(x^{2}e^{x}g+axe^{x}g-xe^{x}g+e^{x}g\right)x=-x^{2}$$

Use the distributive property to multiply x^{2}e^{x}g+axe^{x}g-xe^{x}g+e^{x}g by x.

$$e^{x}gx^{3}+ae^{x}gx^{2}-e^{x}gx^{2}+e^{x}gx=-x^{2}$$

Combine all terms containing g.

$$\left(e^{x}x^{3}+ae^{x}x^{2}-e^{x}x^{2}+e^{x}x\right)g=-x^{2}$$

The equation is in standard form.

$$\left(x^{3}e^{x}+ax^{2}e^{x}-x^{2}e^{x}+xe^{x}\right)g=-x^{2}$$

Divide both sides by e^{x}x^{3}+ae^{x}x^{2}-e^{x}x^{2}+e^{x}x.

$$\frac{\left(x^{3}e^{x}+ax^{2}e^{x}-x^{2}e^{x}+xe^{x}\right)g}{x^{3}e^{x}+ax^{2}e^{x}-x^{2}e^{x}+xe^{x}}=-\frac{x^{2}}{x^{3}e^{x}+ax^{2}e^{x}-x^{2}e^{x}+xe^{x}}$$

Dividing by e^{x}x^{3}+ae^{x}x^{2}-e^{x}x^{2}+e^{x}x undoes the multiplication by e^{x}x^{3}+ae^{x}x^{2}-e^{x}x^{2}+e^{x}x.

$$g=-\frac{x^{2}}{x^{3}e^{x}+ax^{2}e^{x}-x^{2}e^{x}+xe^{x}}$$

Divide -x^{2} by e^{x}x^{3}+ae^{x}x^{2}-e^{x}x^{2}+e^{x}x.

$$g=-\frac{x}{\left(x^{2}+ax-x+1\right)e^{x}}$$