Solve (x^2+bx-2)(x+3)=x^3+6x^2+7x-6

$$( x ^ { 2 } + b x - 2 ) ( x + 3 ) = x ^ { 3 } + 6 x ^ { 2 } + 7 x - 6$$

Solve for b (complex solution)

$$\left\{\begin{matrix}\\b=3\text{, }&\text{unconditionally}\\b\in \mathrm{C}\text{, }&x=-3\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{2}+bx-2 by x+3.

$$x^{3}+3x^{2}+bx^{2}+3bx-2x-6=x^{3}+6x^{2}+7x-6$$

Subtract x^{3} from both sides.

$$3x^{2}+bx^{2}+3bx-2x-6=x^{3}+6x^{2}+7x-6-x^{3}$$

Combine x^{3} and -x^{3} to get 0.

$$3x^{2}+bx^{2}+3bx-2x-6=6x^{2}+7x-6$$

Subtract 3x^{2} from both sides.

$$bx^{2}+3bx-2x-6=6x^{2}+7x-6-3x^{2}$$

Combine 6x^{2} and -3x^{2} to get 3x^{2}.

$$bx^{2}+3bx-2x-6=3x^{2}+7x-6$$

Add 2x to both sides.

$$bx^{2}+3bx-6=3x^{2}+7x-6+2x$$

Combine 7x and 2x to get 9x.

$$bx^{2}+3bx-6=3x^{2}+9x-6$$

Add 6 to both sides.

$$bx^{2}+3bx=3x^{2}+9x-6+6$$

Add -6 and 6 to get 0.

$$bx^{2}+3bx=3x^{2}+9x$$

Combine all terms containing b.

$$\left(x^{2}+3x\right)b=3x^{2}+9x$$

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

$$\frac{\left(x^{2}+3x\right)b}{x^{2}+3x}=\frac{3x\left(x+3\right)}{x^{2}+3x}$$

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

$$b=\frac{3x\left(x+3\right)}{x^{2}+3x}$$

Divide 3x\left(3+x\right) by x^{2}+3x.

$$b=3$$

Solve for b

$$\left\{\begin{matrix}\\b=3\text{, }&\text{unconditionally}\\b\in \mathrm{R}\text{, }&x=-3\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{2}+bx-2 by x+3.

$$x^{3}+3x^{2}+bx^{2}+3bx-2x-6=x^{3}+6x^{2}+7x-6$$

Subtract x^{3} from both sides.

$$3x^{2}+bx^{2}+3bx-2x-6=x^{3}+6x^{2}+7x-6-x^{3}$$

Combine x^{3} and -x^{3} to get 0.

$$3x^{2}+bx^{2}+3bx-2x-6=6x^{2}+7x-6$$

Subtract 3x^{2} from both sides.

$$bx^{2}+3bx-2x-6=6x^{2}+7x-6-3x^{2}$$

Combine 6x^{2} and -3x^{2} to get 3x^{2}.

$$bx^{2}+3bx-2x-6=3x^{2}+7x-6$$

Add 2x to both sides.

$$bx^{2}+3bx-6=3x^{2}+7x-6+2x$$

Combine 7x and 2x to get 9x.

$$bx^{2}+3bx-6=3x^{2}+9x-6$$

Add 6 to both sides.

$$bx^{2}+3bx=3x^{2}+9x-6+6$$

Add -6 and 6 to get 0.

$$bx^{2}+3bx=3x^{2}+9x$$

Combine all terms containing b.

$$\left(x^{2}+3x\right)b=3x^{2}+9x$$

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

$$\frac{\left(x^{2}+3x\right)b}{x^{2}+3x}=\frac{3x\left(x+3\right)}{x^{2}+3x}$$

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

$$b=\frac{3x\left(x+3\right)}{x^{2}+3x}$$

Divide 3x\left(3+x\right) by x^{2}+3x.

$$b=3$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-3\text{; }x=0\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&b=3\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=-3\text{; }x=0\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&b=3\end{matrix}\right.$$