Solve the following equation in the real number system. Please show all of your work.

x^4 + 6x^3 + x^2 - 24x - 20 = 0

Help!!!!

Let f(x)=x^4+6x^3+x^2-24x-20

Real roots are looking for among
+-1,+-2,+-4,+-5,+-10,+-20
f(-1)=1-6+1+24-20=0
Divide f(x) by (x+1):
f(x)=(x+1)(x^3+5x^2-4x-20)
Let g(x)=x^3+5x^2-4x-20
g(-5)=-125+125+20-20=0
Divide g(x) by (x+5):
g(x)=(x+5)(x^2-4)

f(x)=(x+1)(x+5)(x+2)(x-2)

what is 2/5 in the real number system

To solve the equation x^4 + 6x^3 + x^2 - 24x - 20 = 0, follow these steps:

Step 1: Start by looking for any common factors that can be factored out from all the terms. In this case, there are no common factors.

Step 2: If factoring doesn't work, try to use other methods to solve the equation. One such method is the Rational Root Theorem, which can help find possible rational roots. The Rational Root Theorem states that if a rational number p/q is a root of a polynomial equation with integer coefficients, then p must be a factor of the constant term and q must be a factor of the leading coefficient.

In this equation, the constant term is -20, and the leading coefficient is 1. So, the possible rational roots are factors of -20 divided by factors of 1. The factors of -20 are ±1, ±2, ±4, ±5, ±10, and ±20, and the factors of 1 are ±1.

Step 3: Test each of the possible rational roots using synthetic division to see if it gives a remainder of zero.

- Using synthetic division with x = 1, we get the following:
1 │ 1 6 1 -24 -20
│ 1 7 8 -16
--------------------------------
1 7 8 -16 -36

- Since the remainder is not zero, x = 1 is not a root.

- Similarly, we can test other possible rational roots (±2, ±4, ±5, ±10, and ±20). In this case, none of them will yield a remainder of zero.

Step 4: Since the equation does not have any rational roots, we need to resort to numerical methods to find the approximate solutions. One common method is using graphical analysis or a calculator.

- By graphing the equation or using a graphing calculator, we can see that the equation has two real roots. Approximating the graph, we can estimate the roots to be around -3 and 1.5.

So, the approximate solutions to the equation x^4 + 6x^3 + x^2 - 24x - 20 = 0 in the real number system are x ≈ -3 and x ≈ 1.5.