Sure, let's solve this equation step-by-step.
We start with the given equation:
[tex]\[2 \cdot B(x) \cdot C(x) = 2 x^3 - 2 x^5 - 3(4 x^2 + 2 x - 8)\][/tex]
First, let's simplify the expression on the right-hand side. We'll distribute and combine like terms step-by-step.
1. Distribute the [tex]\(-3\)[/tex] through the terms inside the parentheses:
[tex]\[-3(4 x^2 + 2 x - 8)\][/tex]
We get:
[tex]\[-3 \cdot 4 x^2 + (-3) \cdot 2 x + (-3) \cdot (-8) = -12 x^2 - 6 x + 24\][/tex]
2. Now, substitute this back into the original equation:
[tex]\[2 B(x) C(x) = 2 x^3 - 2 x^5 - 12 x^2 - 6 x + 24\][/tex]
3. Combine all the terms on the right-hand side to form one polynomial:
[tex]\[2 B(x) C(x) = -2 x^5 + 2 x^3 - 12 x^2 - 6 x + 24\][/tex]
So, the simplified equation is:
[tex]\[2 B(x) C(x) = -2 x^5 + 2 x^3 - 12 x^2 - 6 x + 24\][/tex]
The simplified right-hand side of the equation is:
[tex]\[-2 x^5 + 2 x^3 - 12 x^2 - 6 x + 24\][/tex]
Thus, the complete step-by-step solution to the given equation is presented as:
[tex]\[2 B(x) C(x) = -2 x^5 + 2 x^3 - 12 x^2 - 6 x + 24\][/tex]
This is the final simplified form of the given equation.
