1. use what you have learned to find the solutions of the polynomial equation shown below. x^4 + 5x^3 ‒ x^2 ‒ 50x ‒ 90 = 0 if x^2 ‒ 10 is one of the factors. be sure to discuss real and complex solutions and how these solutions will affect the graph of the polynomial equation. 2. use what you have learned in this lesson to determine all of the values for k that would give complex solutions to the equation shown below. 3x^2 ‒ 8x + k = 0 describe how you know your values for k give complex solutions to the equation. then, write a quadratic equation using a value of k that will have complex solutions and solve.

If (x^2 -10) is one of the factors, that can be further factored into:
(x - sqrt(10) ) * (x+sqrt(10)) =0
making 2 of the 4 solutions equal:
3.1623 and -3.1623

I then used an algebraic long division calculator
http://calculus-calculator.com/longdivision/
to calculate:
x^4 + 5x^3 ‒ x^2 ‒ 50x ‒ 90 divided by x^2 -10 which equals

x^2 + 5x + 9

Using the quadratic formula, the roots of that equation are:
x = -5 + sqrt (-11) / 2
and
x = -5 - sqrt (-11) / 2

Both of those roots are not real.

I tried using online graphing calculators for x^4+5x^3-x^2-50x-90=0 but none worked.

2.  For this equation,
3x^2 ‒ 8x + k = 0
I used my OWN quadratic formula calculator
http://www.1728.org/quadratc.htm
and found that real roots no longer exist after "k" is greater than 5.3

Step 1: 6x-10=1; step 2: 6x=11