Solve (x+a)(x^2-2x+5)=x^3-4x^2+9x-10

$$( x + a ) ( x ^ { 2 } - 2 x + 5 ) = x ^ { 3 } - 4 x ^ { 2 } + 9 x - 10$$

Solve for a (complex solution)

$$\left\{\begin{matrix}\\a=-2\text{, }&\text{unconditionally}\\a\in \mathrm{C}\text{, }&x=1+2i\text{ or }x=1-2i\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+a by x^{2}-2x+5.

$$x^{3}-2x^{2}+5x+ax^{2}-2ax+5a=x^{3}-4x^{2}+9x-10$$

Subtract x^{3} from both sides.

$$-2x^{2}+5x+ax^{2}-2ax+5a=x^{3}-4x^{2}+9x-10-x^{3}$$

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

$$-2x^{2}+5x+ax^{2}-2ax+5a=-4x^{2}+9x-10$$

Add 2x^{2} to both sides.

$$5x+ax^{2}-2ax+5a=-4x^{2}+9x-10+2x^{2}$$

Combine -4x^{2} and 2x^{2} to get -2x^{2}.

$$5x+ax^{2}-2ax+5a=-2x^{2}+9x-10$$

Subtract 5x from both sides.

$$ax^{2}-2ax+5a=-2x^{2}+9x-10-5x$$

Combine 9x and -5x to get 4x.

$$ax^{2}-2ax+5a=-2x^{2}+4x-10$$

Combine all terms containing a.

$$\left(x^{2}-2x+5\right)a=-2x^{2}+4x-10$$

Divide both sides by x^{2}-2x+5.

$$\frac{\left(x^{2}-2x+5\right)a}{x^{2}-2x+5}=\frac{-2x^{2}+4x-10}{x^{2}-2x+5}$$

Dividing by x^{2}-2x+5 undoes the multiplication by x^{2}-2x+5.

$$a=\frac{-2x^{2}+4x-10}{x^{2}-2x+5}$$

Divide -2x^{2}+4x-10 by x^{2}-2x+5.

$$a=-2$$

Solve for a

$$a=-2$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+a by x^{2}-2x+5.

$$x^{3}-2x^{2}+5x+ax^{2}-2ax+5a=x^{3}-4x^{2}+9x-10$$

Subtract x^{3} from both sides.

$$-2x^{2}+5x+ax^{2}-2ax+5a=x^{3}-4x^{2}+9x-10-x^{3}$$

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

$$-2x^{2}+5x+ax^{2}-2ax+5a=-4x^{2}+9x-10$$

Add 2x^{2} to both sides.

$$5x+ax^{2}-2ax+5a=-4x^{2}+9x-10+2x^{2}$$

Combine -4x^{2} and 2x^{2} to get -2x^{2}.

$$5x+ax^{2}-2ax+5a=-2x^{2}+9x-10$$

Subtract 5x from both sides.

$$ax^{2}-2ax+5a=-2x^{2}+9x-10-5x$$

Combine 9x and -5x to get 4x.

$$ax^{2}-2ax+5a=-2x^{2}+4x-10$$

Combine all terms containing a.

$$\left(x^{2}-2x+5\right)a=-2x^{2}+4x-10$$

Divide both sides by x^{2}-2x+5.

$$\frac{\left(x^{2}-2x+5\right)a}{x^{2}-2x+5}=\frac{-2x^{2}+4x-10}{x^{2}-2x+5}$$

Dividing by x^{2}-2x+5 undoes the multiplication by x^{2}-2x+5.

$$a=\frac{-2x^{2}+4x-10}{x^{2}-2x+5}$$

Divide -2x^{2}+4x-10 by x^{2}-2x+5.

$$a=-2$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=1+2i\text{; }x=1-2i\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&a=-2\end{matrix}\right.$$

Solve for x

$$x\in \mathrm{R}$$

$$a=-2$$