Solution - Polynomial long division

Step  1  :

Equation at the end of step  1  :

  ((((r4)+(2•(r3)))-11r2)-12r)+36
 Step  2  :

Equation at the end of step  2  :

  ((((r4) +  2r3) -  11r2) -  12r) +  36

Step  3  :

Polynomial Roots Calculator :

 3.1    Find roots (zeroes) of :       F(r) = r⁴+2r³-11r²-12r+36
Polynomial Roots Calculator is a set of methods aimed at finding values of  r  for which   F(r)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  r  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  36.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,4 ,6 ,9 ,12 ,18 ,36

 Let us test ....

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that
   r⁴+2r³-11r²-12r+36 
can be divided by 2 different polynomials,including by  r-2 

Polynomial Long Division :

 3.2    Polynomial Long Division
Dividing :  r⁴+2r³-11r²-12r+36 
                              ("Dividend")
By         :    r-2    ("Divisor")

Quotient :  r³+4r²-3r-18  Remainder:  0 

Polynomial Roots Calculator :

 3.3    Find roots (zeroes) of :       F(r) = r³+4r²-3r-18

     See theory in step 3.1
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -18.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,6 ,9 ,18

 Let us test ....

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that
   r³+4r²-3r-18 
can be divided by 2 different polynomials,including by  r-2 

Polynomial Long Division :

 3.4    Polynomial Long Division
Dividing :  r³+4r²-3r-18 
                              ("Dividend")
By         :    r-2    ("Divisor")

Quotient :  r²+6r+9  Remainder:  0 

Trying to factor by splitting the middle term

 3.5     Factoring  r²+6r+9 

The first term is,  r²  its coefficient is  1 .
The middle term is,  +6r  its coefficient is  6 .
The last term, "the constant", is  +9 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 9 = 9 

Step-2 : Find two factors of  9  whose sum equals the coefficient of the middle term, which is   6 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  3  and  3 
                     r² + 3r + 3r + 9

Step-4 : Add up the first 2 terms, pulling out like factors :
                    r • (r+3)
              Add up the last 2 terms, pulling out common factors :
                    3 • (r+3)
Step-5 : Add up the four terms of step 4 :
                    (r+3)  •  (r+3)
             Which is the desired factorization

Multiplying Exponential Expressions :

 3.6    Multiply  (r+3)  by  (r+3) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (r+3)  and the exponents are :
          1 , as  (r+3)  is the same number as  (r+3)¹ 
 and   1 , as  (r+3)  is the same number as  (r+3)¹ 
The product is therefore,  (r+3)⁽¹⁺¹⁾ = (r+3)² 

Multiplying Exponential Expressions :

 3.7    Multiply  (r-2)  by  (r-2) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (r-2)  and the exponents are :
          1 , as  (r-2)  is the same number as  (r-2)¹ 
 and   1 , as  (r-2)  is the same number as  (r-2)¹ 
The product is therefore,  (r-2)⁽¹⁺¹⁾ = (r-2)² 

Final result :

  (r + 3)2 • (r - 2)2