## Solving the Equation: (x+1)(2x+3)(2x+5)(x+3) = 945

This equation presents a challenge due to its high degree and the presence of multiple factors. Let's break down the steps to solve it:

### 1. Expand and Simplify

First, we need to expand the product on the left-hand side of the equation. This can be done systematically:

**Step 1:**Expand the first two factors: (x+1)(2x+3) = 2x² + 5x + 3**Step 2:**Expand the last two factors: (2x+5)(x+3) = 2x² + 11x + 15**Step 3:**Now we need to multiply the two expanded expressions: (2x² + 5x + 3)(2x² + 11x + 15) = 4x⁴ + 22x³ + 30x² + 10x³ + 55x² + 75x + 6x² + 33x + 45**Step 4:**Combine like terms: 4x⁴ + 32x³ + 91x² + 108x + 45 = 945

### 2. Rearrange and Factor

Now we have a fourth-degree polynomial equation. To solve this, we need to rearrange it and attempt to factor it:

**Step 1:**Subtract 945 from both sides: 4x⁴ + 32x³ + 91x² + 108x - 900 = 0**Step 2:**Look for common factors: 4(x⁴ + 8x³ + 22.75x² + 27x - 225) = 0

At this point, factoring the polynomial directly becomes difficult. It's likely that this equation has **rational roots**, which are roots that can be expressed as fractions.

### 3. Finding Rational Roots (Rational Root Theorem)

The Rational Root Theorem helps us find potential rational roots:

**Step 1:**Identify the constant term ( -225) and the leading coefficient (1) of the polynomial.**Step 2:**List all the factors of the constant term: ±1, ±3, ±5, ±9, ±15, ±25, ±45, ±75, ±150, ±225.**Step 3:**List all the factors of the leading coefficient: ±1.**Step 4:**Form all possible fractions by dividing each factor of the constant term by each factor of the leading coefficient: ±1, ±3, ±5, ±9, ±15, ±25, ±45, ±75, ±150, ±225.

Now, we need to test these potential rational roots by substituting them into the equation. We find that **x = 3** is a root, because:

4(3⁴ + 8(3³) + 22.75(3²) + 27(3) - 225) = 0

### 4. Factor with the Found Root

Since x = 3 is a root, (x - 3) is a factor of the polynomial. We can use polynomial long division or synthetic division to find the remaining factors:

**Step 1:**Divide (x⁴ + 8x³ + 22.75x² + 27x - 225) by (x - 3).**Step 2:**You will obtain the quotient: x³ + 11x² + 55.75x + 75.

Therefore, we can rewrite the equation as:

4(x - 3)(x³ + 11x² + 55.75x + 75) = 0

### 5. Finding Remaining Roots

Now, we need to find the roots of the cubic polynomial (x³ + 11x² + 55.75x + 75). This can be a challenge to factor directly. We can use numerical methods or graphing calculators to find the remaining roots.

The remaining roots are approximately:

- x ≈ -5.5
- x ≈ -2.5

### 6. Solutions

Therefore, the solutions to the equation (x+1)(2x+3)(2x+5)(x+3) = 945 are:

- x = 3
- x ≈ -5.5
- x ≈ -2.5