Simplify \( \left(2 x^{3}+8 x^{2}+4 x\right)+\left(6 x^{3}+3 x^{2}-6 x\right) \) \( 8 x^{3}+11 x^{2}+2 x \) \( 8 x^{3}+11 x^{2}+10 x \) \( 8 x^{3}+11 x^{2}-2 x \) \( 8 x^{3}+5 x^{2}-2 x \)

To simplify \( \left(2 x^{3}+8 x^{2}+4 x\right)+\left(6 x^{3}+3 x^{2}-6 x\right) \), combine the like terms. First, for the \(x^3\) terms: \(2x^3 + 6x^3 = 8x^3\). Next, for the \(x^2\) terms: \(8x^2 + 3x^2 = 11x^2\). Finally, for the \(x\) terms: \(4x - 6x = -2x\). Putting it all together, we get: \[ 8x^3 + 11x^2 - 2x. \] So, the correct answer is: \( 8 x^{3}+11 x^{2}-2 x \). Now let's explore the world of polynomials! Polynomials were first introduced in ancient times, with early forms of algebra recorded in Egypt and Babylon. In the Middle Ages, European mathematicians began formalizing these expressions. They have since evolved into crucial tools for modeling real-world phenomena, from physics to economics, proving the versatility and importance of these mathematical constructs! In real-world applications, polynomials are widely used in fields like physics for motion equations, in computer graphics for curve design, and even in finance for modeling growth. Their flexibility allows for complex modeling while remaining mathematically manageable, which is why you see them pop up everywhere!