Question 1208646
<font color=black size=3>
Answer is  <font color=red>f(x) = x^5 - 6x^4 - 13x^3 + 98x^2 - 14x + 104</font>


How to find that answer:


Since all of the coefficients are real numbers, this means the complex roots come in conjugate pairs.
a+bi pairs with a-bi


-i pairs with i
5+i pairs with 5-i


The five roots are: -4, i, -i, 5+i, 5-i
Recall the fundamental theorem of algebra says that any n<sup>th</sup> degree polynomial has n complex roots.


x = i has both sides square to x^2 = -1 and then we get everything to one side: x^2+1=0. So that yields the factor (x^2+1).


x = 5+i becomes x-5 = i; then both sides square to x^2-10x+25 = -1 and it becomes x^2-10x+26=0
Use the quadratic formula to solve x^2-10x+26=0 and you should get x = 5+i and x = 5-i.
Note: If you're using GeoGebra, you need to use the <a href="https://geogebra.github.io/docs/manual/en/commands/CSolve/">CSolve</a> command (in contrast to the regular <a href="https://geogebra.github.io/docs/manual/en/commands/Solve/">Solve</a> command). Otherwise, it will produce an empty set of solutions.


That slight tangent aside, we can say the following
The root x = -4 leads to the factor (x+4)
The roots x = i, x = -i lead to the factor (x^2+1)
The roots x = 5+i, x = 5-i lead to the factor (x^2-10x+26)


The goal is to expand this out:  (x+4)(x^2+1)(x^2-10x+26)


--------------------------------------------------------------------------


For now let's focus on expanding the portion (x+4)(x^2+1)


We could use the FOIL rule, but I'll use the <a href="https://www.algebra.com/algebra/homework/playground/lessons/box-method.lesson">box method</a> instead.
Place the terms x and 4 along the left hand side. Place the terms x^2 and 1 along the top.
<table border = "1" cellpadding = "5"><tr><td></td><td>x^2</td><td>1</td></tr><tr><td>x</td><td></td><td></td></tr><tr><td>4</td><td></td><td></td></tr></table>


To fill out this table, multiply each pair of headers. 
Eg: top left corner is x^2*x = <font color=blue>x^3</font>
<table border = "1" cellpadding = "5"><tr><td></td><td>x^2</td><td>1</td></tr><tr><td>x</td><td><font color=blue>x^3</font></td><td><font color=blue>x</font></td></tr><tr><td>4</td><td><font color=blue>4x^2</font></td><td><font color=blue>4</font></td></tr></table>
The inner terms in <font color=blue>blue</font> are then added to get x^3+4x^2+x+4. There aren't any like terms to combine.


We have shown that (x+4)(x^2+1) = x^3+4x^2+x+4


So 
(x+4)(x^2+1)(x^2-10x+26)
updates to
(x^3+4x^2+x+4)(x^2-10x+26)


We'll need to do one more application of the box method.
Here's the blank template with the headers filled in.
<table border = "1" cellpadding = "5"><tr><td></td><td>x^2</td><td>-10x</td><td>26</td></tr><tr><td>x^3</td><td></td><td></td><td></td></tr><tr><td>4x^2</td><td></td><td></td><td></td></tr><tr><td>x</td><td></td><td></td><td></td></tr><tr><td>4</td><td></td><td></td><td></td></tr></table>


And here's the completed table
<table border = "1" cellpadding = "5"><tr><td></td><td>x^2</td><td>-10x</td><td>26</td></tr><tr><td>x^3</td><td><font color=blue>x^5</font></td><td><font color=blue>-10x^4</font></td><td><font color=blue>26x^3</font></td></tr><tr><td>4x^2</td><td><font color=blue>4x^4</font></td><td><font color=blue>-40x^3</font></td><td><font color=blue>104x^2</font></td></tr><tr><td>x</td><td><font color=blue>x^3</font></td><td><font color=blue>-10x^2</font></td><td><font color=blue>26x</font></td></tr><tr><td>4</td><td><font color=blue>4x^2</font></td><td><font color=blue>-40x</font></td><td><font color=blue>104</font></td></tr></table>
Add up the terms in blue. 
This time we have groups of like terms to combine (eg: 4x^4 + (-10x^4) = -6x^4)
Notice the like terms are along northeast diagonals. 
I'll skip a bit of scratch work and leave it to the student. 


You should get the final result <font color=red>f(x) = x^5 - 6x^4 - 13x^3 + 98x^2 - 14x + 104</font>


You can use various software tools to verify this answer.
<a href="https://www.wolframalpha.com/input?i=x%5E5+-+6x%5E4+-+13x%5E3+%2B+98x%5E2+-+14x+%2B+104%3D0">WolframAlpha</a> is one example. <a href="https://www.geogebra.org/">GeoGebra</a> is another (make sure to use Csolve instead of Solve).
</font>