Part A. The total lenght of the two sides is given the sum of the given sides:
[tex](3x^2-4x-1)+(4x-x^2+5)[/tex]
By combining similar terms, we get
[tex]3x^2-x^2-4x+4x-1+5[/tex]
which gives
[tex]2x^2+4[/tex]
Then, the answer for Part A is
[tex]2x^2+4[/tex]
Part B. We know that the perimeter is given by
[tex]\text{side}1+\text{side}2+\text{side}3=\text{Perimeter}[/tex]
From the previous result, we know that
[tex]\text{side}1+\text{side}2=2x^2+4[/tex]
Then, by subsituting this result into the perimeter formula, we get
[tex]2x^2+4+side3=5x^3-2x^2+3x-8[/tex]
and we need to isolate side3. Then, by moving 2x^2+4 to the right hand side, we get
[tex]\text{side}3=5x^3-2x^2+3x-8-2x^2-4[/tex]
By combining similar terms, we have
[tex]\text{side}3=5x^3-2x^2-2x^2+3x-8-4[/tex]
which gives
[tex]\text{side}3=5x^3-4x^2+3x-12[/tex]
So, the answer for Part B is
[tex]\text{side}3=5x^3-4x^2+3x-12[/tex]
Part C.
In math, a set is closer under addition or substraction if performing that operations on the members of the set always produces a member of that set. Something similar occur to integers, namely, they are closed under addition or subtraction because if you add 2 intergers, the result is another integer.
Then, the answer is yes, polynomials are closed under addition and subtraction because if we add 2 the members always produces another polynomial.
