Factoring and Expanding (x+3)(x+1)(x4)
This expression represents the product of three binomials: (x+3), (x+1), and (x4). We can explore its properties by expanding and factoring it.
Expanding the Expression
To expand the expression, we can use the distributive property (sometimes called FOIL) multiple times.

Start with the first two binomials: (x+3)(x+1) = x² + x + 3x + 3 = x² + 4x + 3

Multiply the result by the third binomial: (x² + 4x + 3)(x4) = x³  4x² + 4x²  16x + 3x  12 = x³  13x  12
Therefore, the expanded form of (x+3)(x+1)(x4) is x³  13x  12.
Factoring the Expression
We can also factor the expression by reversing the expansion process. Here's how:

Identify the roots: The roots of the expression are the values of x that make the expression equal to zero. We can find the roots by setting the expression equal to zero and solving for x: x³  13x  12 = 0 This can be factored as: (x+3)(x+1)(x4) = 0 Therefore, the roots are x = 3, x = 1, and x = 4.

Construct the factored form: Since we know the roots, we can write the factored form as: (x+3)(x+1)(x4)
Conclusion
The expression (x+3)(x+1)(x4) represents a cubic polynomial. By expanding the expression, we get x³  13x  12. By factoring the expression, we can identify its roots and express it as (x+3)(x+1)(x4). Understanding how to expand and factor expressions like this is crucial in algebra, as it allows us to manipulate and solve equations.