(x+1)(x+2)(x+3)(x+6)-3x^2 Solution

3 min read Jun 16, 2024

Solving the Expression (x+1)(x+2)(x+3)(x+6) - 3x^2

This expression involves expanding multiple factors and then simplifying the result. Let's break down the steps:

First, we need to expand the product of the four factors:

(x+1)(x+2)(x+3)(x+6)

We can do this systematically, expanding two factors at a time:

Step 1: Expand (x+1)(x+2) = x² + 3x + 2
Step 2: Expand (x+3)(x+6) = x² + 9x + 18
Step 3: Now we have (x² + 3x + 2)(x² + 9x + 18). We need to multiply these two quadratics. This can be done using the distributive property or a tabular method.

Using the distributive property:

Multiply each term of the first quadratic by each term of the second quadratic.
This results in a total of 9 terms:
- x²(x²) + x²(9x) + x²(18) + 3x(x²) + 3x(9x) + 3x(18) + 2(x²) + 2(9x) + 2(18)

Simplifying the expression:

Combining like terms, we get:

x⁴ + 12x³ + 47x² + 72x + 36

Now, we can substitute the expanded form back into the original expression:

(x+1)(x+2)(x+3)(x+6) - 3x² = (x⁴ + 12x³ + 47x² + 72x + 36) - 3x²

Finally, we combine the x² terms:

x⁴ + 12x³ + 44x² + 72x + 36

Therefore, the solution to the expression (x+1)(x+2)(x+3)(x+6) - 3x² is x⁴ + 12x³ + 44x² + 72x + 36

This expression cannot be factored further using simple techniques. It represents a polynomial function of degree 4.