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Step by Step Solution

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

y-((4*x^4+7*x^2+4)/(2*x^2+x+2))=0

Step 1 :

Equation at the end of step 1 :

    (((4•(x⁴))+(7•(x²)))+4)
  y-———————————————————————  = 0 
          ((2x²+x)+2)      
 Step  2  :

Equation at the end of step 2 :

    (((4•(x⁴))+7x²)+4)
  y-——————————————————  = 0 
        (2x²+x+2)     
 Step  3  :

Equation at the end of step 3 :

       ((2²x⁴ + 7x²) + 4)
  y -  ——————————————————  = 0 
         (2x² + x + 2)

Step 4 :

            4x⁴ + 7x² + 4
 Simplify   —————————————
             2x² + x + 2

Trying to factor by splitting the middle term

4.1 Factoring 4x⁴ + 7x² + 4

The first term is, 4x⁴ its coefficient is 4 .
The middle term is, +7x² its coefficient is 7 .
The last term, "the constant", is +4

Step-1 : Multiply the coefficient of the first term by the constant 4 • 4 = 16

Step-2 : Find two factors of 16 whose sum equals the coefficient of the middle term, which is 7 .

-16	+	-1	=	-17
-8	+	-2	=	-10
-4	+	-4	=	-8
-2	+	-8	=	-10
-1	+	-16	=	-17
1	+	16	=	17
2	+	8	=	10
4	+	4	=	8
8	+	2	=	10
16	+	1	=	17

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Trying to factor by splitting the middle term

4.2 Factoring 2x²+x+2

The first term is, 2x² its coefficient is 2 .
The middle term is, +x its coefficient is 1 .
The last term, "the constant", is +2

Step-1 : Multiply the coefficient of the first term by the constant 2 • 2 = 4

Step-2 : Find two factors of 4 whose sum equals the coefficient of the middle term, which is 1 .

-4	+	-1	=	-5
-2	+	-2	=	-4
-1	+	-4	=	-5
1	+	4	=	5
2	+	2	=	4
4	+	1	=	5

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Polynomial Long Division :

4.3    Polynomial Long Division
Dividing : 4x⁴+7x²+4
                              ("Dividend")
By         :    2x²+x+2    ("Divisor")

dividend		4x⁴			+	7x²			+	4
- divisor	* 2x²	4x⁴	+	2x³	+	4x²
remainder			-	2x³	+	3x²			+	4
- divisor	* -x¹		-	2x³	-	x²	-	2x
remainder						4x²	+	2x	+	4
- divisor	* 2x⁰					4x²	+	2x	+	4
remainder										0

Quotient : 2x²-x+2
Remainder : 0

Equation at the end of step 4 :

  y -  (2x² - x + 2)  = 0

Step 5 :

Equation at the end of step 5 :

  y - 2x² + x - 2  = 0

Step 6 :

Solving a Single Variable Equation :

6.1 Solve y-2x²+x-2 = 0

In this type of equations, having more than one variable (unknown), you have to specify for which variable you want the equation solved.

We shall not handle this type of equations at this time.