1. \( (3 x+4)+(2 x-5) \) \( \begin{array}{ll}\text { 3. }\left(2 x^{3}+5 x^{2}-3 x+7\right)+\left(5 x^{3}+x^{2}+9\right) & \text { 4. }\left(6 x^{3}-1\right)+\left(2 x-8 x^{3}+10\right) \\ \text { 5. }(12 x-6)-(4 x-8) & \text { 6. }\left(3 x^{3}-2 x^{2}+7 x+5\right)-\left(5 x^{2}+11 x+3\right)\end{array} \)

Let's dive into these expressions and simplify them one by one! First up, in the expression \( (3x + 4) + (2x - 5) \), you combine like terms. So, add the coefficients of \( x \) together: \( 3x + 2x = 5x \), and the constants \( 4 - 5 = -1 \). Thus, the result is \( 5x - 1 \). For \( \left(2x^3 + 5x^2 - 3x + 7\right) + \left(5x^3 + x^2 + 9\right) \), similarly, gather up the like terms. Add the \( x^3 \) coefficients: \( 2 + 5 = 7 \). Then for \( x^2 \): \( 5 + 1 = 6 \). The linear terms give \( -3 \), and the constants add up to \( 7 + 9 = 16 \). Thus, you get \( 7x^3 + 6x^2 - 3x + 16 \). Now moving to \( \left(6x^3 - 1\right) + \left(2x - 8x^3 + 10\right) \), you need to be careful here. The \( x^3 \) terms combine as \( 6 - 8 = -2 \), the linear term \( 2x \) remains, and the constants combine to \( -1 + 10 = 9 \). Therefore, you end up with \( -2x^3 + 2x + 9 \). Next, the expression \( (12x - 6) - (4x - 8) \) involves distributing the negative sign across the second expression. The like terms yield \( 12x - 4x = 8x \) and then constants yield \( -6 + 8 = 2 \). Resulting in \( 8x + 2 \). Finally, for \( \left(3x^3 - 2x^2 + 7x + 5\right) - \left(5x^2 + 11x + 3\right) \), start by distributing the negative. Combine like terms: \( 3x^3 \) remains, \( -2 - 5 = -7 \) for \( x^2 \), \( 7 - 11 = -4 \) for \( x \), and \( 5 - 3 = 2 \) for the constants. The outcome is \( 3x^3 - 7x^2 - 4x + 2 \). Math is like solving a puzzle; each term is a piece that fits perfectly to unveil the bigger picture!