## Solving the Equation (vi) (x+2)(3x+4)(3x+7)(x+3)=2400

This problem involves solving a fourth-degree polynomial equation. Here's a step-by-step approach to find the solutions:

**1. Expand the Equation**

Begin by expanding the left side of the equation:

- First, multiply the first two factors and the last two factors:
- (x+2)(3x+4) = 3x² + 10x + 8
- (3x+7)(x+3) = 3x² + 16x + 21

- Then multiply the resulting expressions:
- (3x² + 10x + 8)(3x² + 16x + 21) = 9x⁴ + 98x³ + 289x² + 272x + 168

Now the equation becomes: 9x⁴ + 98x³ + 289x² + 272x + 168 = 2400

**2. Rearrange the Equation**

Subtract 2400 from both sides to set the equation to zero:

9x⁴ + 98x³ + 289x² + 272x - 2232 = 0

**3. Finding the Solutions**

Unfortunately, there's no straightforward algebraic method to solve a fourth-degree polynomial equation. Here are common approaches:

**Factoring:**Try to factor the equation. This might be possible if there are rational roots. However, it's often difficult to factor higher-degree polynomials directly.**Rational Root Theorem:**This theorem helps find potential rational roots. If a rational number p/q is a root, then p must be a factor of the constant term (-2232) and q must be a factor of the leading coefficient (9).**Numerical Methods:**Use numerical methods like the Newton-Raphson method or graphing calculators to approximate the solutions.

**4. Using Numerical Methods (Example with Newton-Raphson)**

The Newton-Raphson method is an iterative process. To use it, you need:

**The function:**f(x) = 9x⁴ + 98x³ + 289x² + 272x - 2232**Its derivative:**f'(x) = 36x³ + 294x² + 578x + 272

**Steps:**

**Choose an initial guess (x₀).**You might try a value close to where the graph of the function crosses the x-axis.**Iterate using the formula:**xₙ₊₁ = xₙ - f(xₙ) / f'(xₙ)**Repeat step 2 until the value of x converges to a solution.**

**Note:** The Newton-Raphson method may not always converge, and you might need to try different initial guesses.

**Conclusion**

Solving the equation (vi) (x+2)(3x+4)(3x+7)(x+3)=2400 involves expanding the equation, rearranging it, and then using numerical methods or potentially factoring to find the solutions. While factoring can be challenging for higher-degree polynomials, numerical methods like the Newton-Raphson method offer a robust way to approximate the solutions.