Answer
The sum of two polynomials of equal degree (say that the degree is $k$) has a degree that is less than or equal to $k$
Work Step by Step
When adding two polynomials together, you add the coefficients in front of the powers of the variable. For example:
$(x^{2} + 2x + 1) + (3x^{2} - x + 4) = (1+3)x^{2} + (2 -1)x + (1+4) $
$= 4x^{2} + x +5$
As such, the addition has no effect on the powers of your variable, $x$ for example, in the polynomial. The addition only affects the coefficients in front of the powers of the variable, so addition of two polynomials of the same degree cannot produce a polynomial of higher than their degrees. Though, addition of two polynomials can decrease the overall degree in the case where the coefficient of the highest degree term in one of the polynomials is the negative of the the coefficient of the highest degree term in the other polynomial.
This is illustrated in the example below:
$(x^{2} + 2x + 1) + (-x^{2} - x + 4) = (1-1)x^{2} + (2 -1)x + (1+4) $
$= x +5$
