Solve f(x)^prime=2te^x(x+1)

$$f ( x ) ^ { \prime } = 2 t e ^ { x } ( x + 1 )$$

Solve for t (complex solution)

$$\left\{\begin{matrix}t=\frac{f}{2\left(x+1\right)e^{x}}\text{, }&x\neq -1\\t\in \mathrm{C}\text{, }&f=0\text{ and }x=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2te^{x} by x+1.

$$f\frac{\mathrm{d}}{\mathrm{d}x}(x)=2te^{x}x+2te^{x}$$

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

$$2te^{x}x+2te^{x}=f\frac{\mathrm{d}}{\mathrm{d}x}(x)$$

Combine all terms containing t.

$$\left(2e^{x}x+2e^{x}\right)t=f\frac{\mathrm{d}}{\mathrm{d}x}(x)$$

The equation is in standard form.

$$\left(2xe^{x}+2e^{x}\right)t=f$$

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

$$\frac{\left(2xe^{x}+2e^{x}\right)t}{2xe^{x}+2e^{x}}=\frac{f}{2xe^{x}+2e^{x}}$$

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

$$t=\frac{f}{2xe^{x}+2e^{x}}$$

Divide f by 2e^{x}x+2e^{x}.

$$t=\frac{f}{2\left(x+1\right)e^{x}}$$

Solve for t

$$\left\{\begin{matrix}t=\frac{f}{2\left(x+1\right)e^{x}}\text{, }&x\neq -1\\t\in \mathrm{R}\text{, }&f=0\text{ and }x=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2te^{x} by x+1.

$$f\frac{\mathrm{d}}{\mathrm{d}x}(x)=2te^{x}x+2te^{x}$$

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

$$2te^{x}x+2te^{x}=f\frac{\mathrm{d}}{\mathrm{d}x}(x)$$

Combine all terms containing t.

$$\left(2e^{x}x+2e^{x}\right)t=f\frac{\mathrm{d}}{\mathrm{d}x}(x)$$

The equation is in standard form.

$$\left(2xe^{x}+2e^{x}\right)t=f$$

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

$$\frac{\left(2xe^{x}+2e^{x}\right)t}{2xe^{x}+2e^{x}}=\frac{f}{2xe^{x}+2e^{x}}$$

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

$$t=\frac{f}{2xe^{x}+2e^{x}}$$

Divide f by 2e^{x}x+2e^{x}.

$$t=\frac{f}{2\left(x+1\right)e^{x}}$$

Solve for f

$$f=2t\left(x+1\right)e^{x}$$