Solution - Polynomial long division

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x2"   was replaced by   "x^2".  2 more similar replacement(s).

Step  1  :

Equation at the end of step  1  :

  ((((0-(3•(x4)))-(8•(x3)))-(22•3x2))-12x)-5
 Step  2  :

Equation at the end of step  2  :

  ((((0-(3•(x4)))-23x3)-(22•3x2))-12x)-5
 Step  3  :

Equation at the end of step  3  :

  ((((0 -  3x4) -  23x3) -  (22•3x2)) -  12x) -  5

Step  4  :

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   -3x⁴ - 8x³ - 12x² - 12x - 5  = 

  -1 • (3x⁴ + 8x³ + 12x² + 12x + 5) 

Polynomial Roots Calculator :

 5.2    Find roots (zeroes) of :       F(x) = 3x⁴ + 8x³ + 12x² + 12x + 5
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  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  3  and the Trailing Constant is  5.

 The factor(s) are:

of the Leading Coefficient :  1,3
 of the Trailing Constant :  1 ,5

 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
   3x⁴ + 8x³ + 12x² + 12x + 5 
can be divided with  x + 1 

Polynomial Long Division :

 5.3    Polynomial Long Division
Dividing :  3x⁴ + 8x³ + 12x² + 12x + 5 
                              ("Dividend")
By         :    x + 1    ("Divisor")

Quotient :  3x³+5x²+7x+5  Remainder:  0 

Polynomial Roots Calculator :

 5.4    Find roots (zeroes) of :       F(x) = 3x³+5x²+7x+5

     See theory in step 5.2
In this case, the Leading Coefficient is  3  and the Trailing Constant is  5.

 The factor(s) are:

of the Leading Coefficient :  1,3
 of the Trailing Constant :  1 ,5

 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
   3x³+5x²+7x+5 
can be divided with  x+1 

Polynomial Long Division :

 5.5    Polynomial Long Division
Dividing :  3x³+5x²+7x+5 
                              ("Dividend")
By         :    x+1    ("Divisor")

Quotient :  3x²+2x+5  Remainder:  0 

Trying to factor by splitting the middle term

 5.6     Factoring  3x²+2x+5 

The first term is,  3x²  its coefficient is  3 .
The middle term is,  +2x  its coefficient is  2 .
The last term, "the constant", is  +5 

Step-1 : Multiply the coefficient of the first term by the constant   3 • 5 = 15 

Step-2 : Find two factors of  15  whose sum equals the coefficient of the middle term, which is   2 .

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

Multiplying Exponential Expressions :

 5.7    Multiply  (x+1)  by  (x+1) 

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

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

Final result :

  (-3x2 - 2x - 5) • (x + 1)2