Answer:
No.
Step-by-step explanation:
No, one easy way to see it is with quadratic formulas. There exists quadratic polynomials with no real solutions, then if you add, subtract or multiply two polynomials and obtain a quadratic formula, possibly this polynomial won't have real solutions.
I am going to give one counterexample:
We have the two polynomials and , then is we subtract q(x)-p(x) we obtain
The resulting polynomial is a quadratic polynomial of the form with a=1, b=1 and c=1. This polynomial has no real solutions, you can check it with the discriminating As the discriminating is negative, the polynomial has no real solutions.