### Instruction: Completely factor the following expression: \[(15x^3+80x-5)-(-4x^3+4x-5).\].### Response: The first thing I notice is that there are two expressions in parentheses, and they are subtracted from each other.
So I can rewrite this as a single expression by distributing the negative sign to the second one: \[15x^3+80x-5+4x^3-4x+5.\]
Next, I can combine the like terms to simplify the expression: \[19x^3+76x.\]
Now I need to factor this expression completely.
I look for the greatest common factor of both terms, which is 19x.
I can factor that out and get: \[19x(x^2+4).\]
I'm not done yet, because I need to check if the expression inside the parentheses can be further factored.
I recognize that it is a sum of squares, which is not factorable over the real numbers, but it is factorable over the complex numbers using the formula: \[a^2+b^2=(a+bi)(a-bi).\]
So I can apply that formula with a=x and b=2, and get: \[x^2+4=(x+2i)(x-2i).\]
Therefore, the final answer is: \[19x(x+2i)(x-2i).\]