1. (20 pts.) Graphically solve the equation and inequality below. Sketch the graph(s) that you used and round your answers to two decimal places (hundredths place). (a) \( 6 x^{2}-2 x^{3}=-x^{2}+5 x+1 \)

Solve the equation by following steps: - step0: Solve for \(x\): \(6x^{2}-2x^{3}=-x^{2}+5x+1\) - step1: Move the expression to the left side: \(6x^{2}-2x^{3}-\left(-x^{2}+5x+1\right)=0\) - step2: Subtract the terms: \(7x^{2}-2x^{3}-5x-1=0\) - step3: Calculate: \(\begin{align}&x\approx -0.161702\\&x\approx 1.321037\\&x\approx 2.340665\end{align}\) - step4: Rewrite: \(x_{1}\approx -0.161702,x_{2}\approx 1.321037,x_{3}\approx 2.340665\) To solve the equation \( 6 x^{2}-2 x^{3}=-x^{2}+5 x+1 \), we first rearrange it into standard form: \[ -2x^3 + 6x^2 + x^2 - 5x - 1 = 0 \] This simplifies to: \[ -2x^3 + 7x^2 - 5x - 1 = 0 \] Next, we can find the roots of this polynomial equation. The solutions are approximately: - \( x_1 \approx -0.16 \) - \( x_2 \approx 1.32 \) - \( x_3 \approx 2.34 \) ### Graphical Representation While I cannot provide a graph directly, you can sketch the graph of the function: \[ f(x) = -2x^3 + 7x^2 - 5x - 1 \] 1. **Plot the function**: You can use graphing software or a graphing calculator to plot \( f(x) \) over the range \( x = -2 \) to \( x = 3 \). 2. **Identify the x-intercepts**: The points where the graph intersects the x-axis correspond to the solutions we found: \( x \approx -0.16, 1.32, 2.34 \). ### Inequality If you need to solve an inequality related to this equation, please specify the inequality, and I can assist you further!