Factoring and Solving the Expression (x+1)(x+3)(x4)(x6) + 24
This article explores the factorization and solution of the expression (x+1)(x+3)(x4)(x6) + 24. We will utilize algebraic techniques to simplify the expression and determine its roots.
Understanding the Expression
The expression (x+1)(x+3)(x4)(x6) + 24 represents the product of four linear factors, each containing a variable 'x', added to a constant term 24. To solve the expression, we need to find the values of 'x' that make the entire expression equal to zero.
Factoring the Expression
The key to simplifying this expression lies in recognizing the pattern within the first part of the expression. Observe that:
 The first two factors (x+1) and (x+3) are similar, with a difference of 2 between the constant terms.
 The last two factors (x4) and (x6) also share a similar structure, with a difference of 2.
This pattern suggests that we can manipulate the expression to form a difference of squares.

Rearrange the factors: (x+1)(x4) * (x+3)(x6) + 24

Expand the first two factors: (x²  3x  4) * (x²  3x  18) + 24

Introduce a new variable: Let y = x²  3x. Substituting y into the expression: (y  4) * (y  18) + 24

Expand and simplify: y²  22y + 72 + 24 = y²  22y + 96

Factor the quadratic: (y  6)(y  16)

Substitute back: (x²  3x  6)(x²  3x  16)
Now the expression is factored into two quadratic factors.
Solving the Expression
To find the roots, we need to set the entire expression equal to zero:
(x²  3x  6)(x²  3x  16) = 0
This equation holds true if either of the quadratic factors is equal to zero. Therefore, we need to solve two separate quadratic equations:

x²  3x  6 = 0
This equation can be solved using the quadratic formula:
x = (b ± √(b²  4ac)) / 2a
where a = 1, b = 3, and c = 6.
Solving for x, we obtain two real roots.

x²  3x  16 = 0
Applying the quadratic formula with a = 1, b = 3, and c = 16, we get two more real roots.
Conclusion
By recognizing the pattern in the expression and using algebraic techniques like substitution and factorization, we were able to simplify the expression and determine its roots. This process highlights the importance of understanding factorization and the quadratic formula in solving complex algebraic expressions.