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

Let's solve each problem step by step by combining like terms. --- **Problem 5** Add the polynomials: \[ (-3a^{3} + 2a^{2} - 4) + (a^{3} - 3a^{2} - 5a + 7) \] **Step-by-Step Solution:** 1. **Combine like terms:** - **\(a^3\) terms:** \(-3a^3 + a^3 = -2a^3\) - **\(a^2\) terms:** \(2a^2 - 3a^2 = -a^2\) - **\(a\) terms:** \(-5a\) (only one term) - **Constant terms:** \(-4 + 7 = 3\) 2. **Write the simplified expression:** \[ -2a^3 - a^2 - 5a + 3 \] --- **Problem 6** Subtract the polynomials: \[ (7x^{2}y^{2} - 6x^{3} + xy) - (5x^{2}y^{2} - x^{3} + xy + x) \] **Step-by-Step Solution:** 1. **Distribute the subtraction across the second polynomial:** \[ 7x^{2}y^{2} - 6x^{3} + xy - 5x^{2}y^{2} + x^{3} - xy - x \] 2. **Combine like terms:** - **\(x^3\) terms:** \(-6x^3 + x^3 = -5x^3\) - **\(x^2y^2\) terms:** \(7x^2y^2 - 5x^2y^2 = 2x^2y^2\) - **\(xy\) terms:** \(xy - xy = 0\) (they cancel out) - **Constant term:** \(-x\) 3. **Write the simplified expression:** \[ -5x^3 + 2x^2y^2 - x \] --- **Final Answers:** 5. \(-2a^3 - a^2 - 5a + 3\) 6. \(-5x^3 + 2x^2 y^2 - x\)