(x+1)(2x+3)(2x+5)(x+3)=945

5 min read Jun 16, 2024

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