Question

solve for x.
$x^3 + 3x^2 - 5x + 25 = 0$
$x = ?, \square \pm \square i$

Explanation:

Step1: Factor by grouping

Group terms and factor common parts:
$$x^3 + 3x^2 -5x +25 = x^2(x+3) -5(x-5)$$
Wait, correct grouping for root finding: use Rational Root Theorem first. Test $x=-5$:
$$(-5)^3 +3(-5)^2 -5(-5)+25 = -125+75+25+25=0$$
So $(x+5)$ is a factor. Perform polynomial division or synthetic division.

Step2: Synthetic division with root -5

Divide $x^3+3x^2-5x+25$ by $(x+5)$:
Using synthetic division:

$$\begin{array}{r|rrrr} -5 & 1 & 3 & -5 & 25 \\ & & -5 & 10 & -25 \\ \hline & 1 & -2 & 5 & 0 \end{array}$$

Resulting quadratic: $x^2-2x+5$

Step3: Solve quadratic equation

Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ for $ax^2+bx+c=0$:
Here $a=1, b=-2, c=5$
$$x=\frac{2\pm\sqrt{(-2)^2-4(1)(5)}}{2(1)}=\frac{2\pm\sqrt{4-20}}{2}=\frac{2\pm\sqrt{-16}}{2}$$
Simplify the radical:
$$x=\frac{2\pm4i}{2}=1\pm2i$$

Answer:

$x=-5$, $x=1+2i$, $x=1-2i$