Solve for x:

2x^3 + 9x^2 +10x +3 =0

By inspection, I see that x = -1 is one solution. Therefore x+1 is a factor.

Divide 2x^3 + 9x^2 +10x +3 by x-1 to get the other factor, a quadratic polynomial.
You should get 2x^2 + 7x + 3, using polynomial long division.

Now set that equal to zero and solve it for the other roots.

It factors easily to (2x+1)(x+3) = 0

The other two roots are x = -1/2 and x = -3

To solve the equation 2x^3 + 9x^2 + 10x + 3 = 0, we can use various methods, such as factoring, the rational root theorem, or numerical methods. Let's start by factoring.

Unfortunately, factoring a cubic polynomial like this one can be quite challenging, especially since there is no apparent common factor among the terms. In such cases, it is common to turn to numerical methods or calculators.

A widely-used numerical approach to solving equations is the Newton-Raphson method, but it requires an initial guess for the solution. Another method is using a graphing calculator or online tools to find the roots of the equation. These tools can provide the exact roots or approximate them, depending on the settings.

Suppose we want to use a graphing calculator or online tool. We can simply rewrite the equation as f(x) = 2x^3 + 9x^2 + 10x + 3 = 0 and plot the graph of f(x). By examining the graph, we can estimate the approximate values of x where the function crosses the x-axis.

If you have access to a graphing calculator or an online tool, enter f(x) = 2x^3 + 9x^2 + 10x + 3 and find the x-values where f(x) = 0.