The equation shown in the image is:

$4x^3 + 8x^2 - 9x - 18 = 0$

This is a cubic equation, which can be solved using techniques such as factoring, the Rational Root Theorem, or synthetic division if possible roots are identified.

Steps to Solve

1. Check for Rational Roots: Using the Rational Root Theorem, possible rational roots would be factors of $-18$ divided by factors of $4$, which gives a set of possible rational roots to test, including $\pm 1$, $\pm 2$, $\pm 3$, $\pm 6$, $\pm 9$, and $\pm 18$, as well as $\pm \frac{1}{2}$, $\pm \frac{3}{2}$, $\pm \frac{9}{2}$, etc.
2. Synthetic Division: Try the possible roots until a root works, and then use synthetic division to reduce the cubic equation to a quadratic equation.
3. Solve the Resulting Quadratic Equation: After obtaining a quadratic from synthetic division, solve it using the quadratic formula if it doesn't factor easily.

Let me know if you would like a detailed walkthrough of each step or assistance with synthetic division and root testing!

Questions to Explore Further

1. How does the Rational Root Theorem help in solving polynomial equations?
2. What are the steps involved in synthetic division for polynomials?
3. How can complex roots affect the solutions of a cubic equation?
4. What are other methods to solve a cubic equation if it does not factor?
5. How does the behavior of a cubic function change with the sign of its leading coefficient?

Tip

When solving cubic equations, always test for simpler integer roots first to reduce the equation before resorting to more complex methods.