Step 1: Simplify the comparison using algebraic transformation
Subtract 1 from both quantities:

Quantity A: \(x^2+x \)
Quantity B: \(y^2+y \)


Consider the difference Quantity A - Quantity B:


\(\text { Quantity } \mathrm{A}-\text { Quantity } \mathrm{B}=\left(x^2+x\right)-\left(y^2+y\right)\)



Step 2: Factor the difference
Group the squared terms and the linear terms together:


\(\text { Quantity } \mathrm{A}-\text { Quantity } \mathrm{B}=\left(x^2-y^2\right)+(x-y)\)



Apply the difference of squares identity \(x^2-y^2=(x-y)(x+y)\) :


\(\text { Quantity A - Quantity } \mathrm{B}=(x-y)(x+y)+(x-y)\)



Factor out (x-y) :


\(\text { Quantity } \mathrm{A}-\text { Quantity } \mathrm{B}=(x-y)(x+y+1)\)


Step 3: Substitute the given condition \(x+y=-1 \)
Substitute \(x+y=-1 \) into the factored expression:


\(\text { Quantity } \mathrm{A}-\text { Quantity } \mathrm{B}=(x-y)(-1+1)=(x-y)(0)=0\)



Since the difference between Quantity A and Quantity B is 0:


\(\text { Quantity } \mathrm{A}=\text { Quantity } \mathrm{B}\)



Conclusion
The two quantities are equal.