1. Add or subtract the polynomials. a. \( \left(4 a^{4}-6 a^{3}-3 a^{2}+a+1\right)+\left(5 a^{3}+7 a^{2}+2 a-2\right) \) b. \( \left(2 a^{2} b^{2}+3 a b^{2}-5 a^{2} b\right)-\left(3 a^{2} b^{2}-9 a^{2} b+7 a b^{2}\right) \)

To add or subtract the polynomials, we combine like terms for each expression. a. Combining the polynomials: \[ (4a^{4} - 6a^{3} - 3a^{2} + a + 1) + (5a^{3} + 7a^{2} + 2a - 2) \] First, combine \(4a^4\) (the only \(a^4\) term), then combine the \(a^3\) terms: \(-6a^3 + 5a^3 = -a^3\). Next, combine the \(a^2\) terms: \(-3a^2 + 7a^2 = 4a^2\). Then add the \(a\) terms: \(a + 2a = 3a\). Lastly, combine the constant terms: \(1 - 2 = -1\). The result is: \[ 4a^4 - a^3 + 4a^2 + 3a - 1 \] b. Subtracting the polynomials: \[ (2a^{2}b^{2} + 3ab^{2} - 5a^{2}b) - (3a^{2}b^{2} - 9a^{2}b + 7ab^{2}) \] Distributing the negative to the second polynomial: \(-(3a^{2}b^{2}) + 9a^{2}b - 7ab^{2}\). Now combine like terms for \(a^2b^2\): \(2a^2b^2 - 3a^2b^2 = -a^2b^2\). Combine the \(a^2b\) terms: \(-5a^2b + 9a^2b = 4a^2b\). Finally, combine the \(ab^2\) terms: \(3ab^2 - 7ab^2 = -4ab^2\). The final result is: \[ -a^{2}b^{2} + 4a^{2}b - 4ab^{2} \]