Let's find the other addend given the sum of two polynomials and one addend.
We have the sum of two polynomials:
[tex]\[10 a^2 b^2 - 8 a^2 b + 6 a b^2 - 4 a b + 2.\][/tex]
One addend is:
[tex]\[-5 a^2 b^2 + 12 a^2 b - 5.\][/tex]
To find the other addend, we subtract the given addend from the sum of the polynomials.
Let [tex]\( P(x) \)[/tex] represent the sum of the polynomials and [tex]\( A(x) \)[/tex] represent the given addend. We can write:
[tex]\[ P(x) = 10 a^2 b^2 - 8 a^2 b + 6 a b^2 - 4 a b + 2 \][/tex]
[tex]\[ A(x) = -5 a^2 b^2 + 12 a^2 b - 5 \][/tex]
Now, let [tex]\( B(x) \)[/tex] represent the other addend. According to the problem:
[tex]\[ P(x) = A(x) + B(x) \][/tex]
To find [tex]\( B(x) \)[/tex], we rearrange the equation:
[tex]\[ B(x) = P(x) - A(x) \][/tex]
Substitute the expressions for [tex]\( P(x) \)[/tex] and [tex]\( A(x) \)[/tex]:
[tex]\[ B(x) = (10 a^2 b^2 - 8 a^2 b + 6 a b^2 - 4 a b + 2) - (-5 a^2 b^2 + 12 a^2 b - 5) \][/tex]
[tex]\[ B(x) = 10 a^2 b^2 - 8 a^2 b + 6 a b^2 - 4 a b + 2 + 5 a^2 b^2 - 12 a^2 b + 5 \][/tex]
Combine the like terms:
[tex]\[ B(x) = (10 a^2 b^2 + 5 a^2 b^2) + (-8 a^2 b - 12 a^2 b) + 6 a b^2 - 4 a b + (2 + 5) \][/tex]
[tex]\[ B(x) = 15 a^2 b^2 - 20 a^2 b + 6 a b^2 - 4 a b + 7 \][/tex]
So, the other addend is:
[tex]\[ 15 a^2 b^2 - 20 a^2 b + 6 a b^2 - 4 a b + 7 \][/tex]
The correct option is:
[tex]\[ 15 a^2 b^2 - 20 a^2 b + 6 a b^2 - 4 a b + 7 \][/tex]
