Answer:
 4p3 + 6p2 - 8p - 7
Step-by-step explanation:
Reformatting the input :
Changes made to your input should not affect the solution:
 (1): "p2"   was replaced by   "p^2".  1 more similar replacement(s).
STEP
1
:
Equation at the end of step 1
  (((4 • (p3)) +  (2•3p2)) -  7) -  8p
 STEP 
2
 :
Equation at the end of step
2
:
  ((22p3 +  (2•3p2)) -  7) -  8p
STEP
3
:
Checking for a perfect cube
 3.1    4p3+6p2-8p-7  is not a perfect cube 
Trying to factor by pulling out :
 3.2      Factoring:  4p3+6p2-8p-7 
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1:  -8p-7 
Group 2:  4p3+6p2 
Pull out from each group separately :
Group 1:   (8p+7) • (-1)
Group 2:   (2p+3) • (2p2)
 3.3    Find roots (zeroes) of :       F(p) = 4p3+6p2-8p-7
Polynomial Roots Calculator is a set of methods aimed at finding values of  p  for which   F(p)=0  
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  p  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  4  and the Trailing Constant is  -7. 
 The factor(s) are: 
of the Leading Coefficient :  1,2 ,4 
 of the Trailing Constant :  1 ,7 
 Let us test ....
  	P    Q    P/Q    F(P/Q)    	Divisor
      -1       1        -1.00        3.00    	
      -1       2        -0.50        -2.00    	
      -1       4        -0.25        -4.69    	
      -7       1        -7.00       -1029.00    	
      -7       2        -3.50        -77.00    	
      -7       4        -1.75        3.94    	
      1       1        1.00        -5.00    	
      1       2        0.50        -9.00    	
      1       4        0.25        -8.56    	
      7       1        7.00        1603.00    	
      7       2        3.50        210.00    	
      7       4        1.75        18.81    	
Final result :
  4p3 + 6p2 - 8p - 7