Please help me with this problem. I don't know what to do.

Use the graph or table to determine a solution of the equation. Use synthetic division to verify that this number is a solution of the equation. Then solve the polynomial equation.

x3 + 6x2 + 11x + 6 = 0

To find the solution of the equation x^3 + 6x^2 + 11x + 6 = 0, we can first look at the graph to get an idea of the possible solutions. However, if you don't have the graph, we can use synthetic division to find a possible solution and then solve the equation.

Let's start by using synthetic division to test for a possible solution.

Step 1: Find a possible root.
We can start by testing integer values as possible roots. Let's try x = -1:

-1 | 1 6 11 6
-1 -5 -6
---------------
1 5 6 0

The result of the synthetic division is 1 5 6 0.

Step 2: Analyze the result.
The synthetic division result gives us a remainder of 0, which means x = -1 is a root/solution of the equation.

Step 3: Solve the equation.
Our equation is x^3 + 6x^2 + 11x + 6 = 0. We know that x = -1 is a root, so we can divide the equation by x + 1:

(x + 1)(x^2 + 5x + 6) = 0

To find the other two roots, we solve the quadratic equation x^2 + 5x + 6 = 0.

(x + 2)(x + 3) = 0

Therefore, the solutions of the polynomial equation x^3 + 6x^2 + 11x + 6 = 0 are x = -1, x = -2, and x = -3.

To find a solution of the equation x^3 + 6x^2 + 11x + 6 = 0 using the graph or table, you can start by plotting the graph of the equation or creating a table of values.

Method 1: Using the Graph
1. Graph the equation y = x^3 + 6x^2 + 11x + 6.
2. Look for the x-intercepts of the graph. These are the points where the curve intersects the x-axis.
3. The x-coordinate of each x-intercept is a solution to the equation.

Method 2: Using a Table of Values
1. Create a table with columns for x and y.
2. Choose a range of x-values that you think will include the solutions to the equation.
3. Substitute each x-value into the equation and calculate the corresponding y-value.
4. Look for x-values where the corresponding y-value is zero. These x-values are solutions to the equation.

Once you have found a possible solution using the graph or table, you can verify it by performing synthetic division.

Synthetic division steps:
1. Write the equation in descending order of degrees (highest to lowest), filling in any missing terms with zeros. In this case, the equation is x^3 + 6x^2 + 11x + 6 = 0.
2. Choose the possible solution you found and write it as the divisor. For example, if you found x = -2 as a possible solution, write (x + 2) as the divisor.
3. Perform synthetic division using the possible solution as the divisor. Write the coefficients of the equation as the dividend. For example, the coefficients of x^3 + 6x^2 + 11x + 6 are 1, 6, 11, and 6.
4. Perform the synthetic division step by step to get the quotient and remainder.
5. If the remainder is zero, the possible solution you chose is indeed a solution of the equation.

Finally, to solve the polynomial equation completely, repeat the above steps with any remaining factors of the original polynomial equation until you no longer have any cubic factors.