To simplify the expression [tex]\(-3 x^3 (x^2 - 2x - 4)\)[/tex], let's proceed step-by-step through the process.
1. Identify the expression to be simplified:
[tex]\[-3 x^3 (x^2 - 2x - 4)\][/tex]
2. Distribute [tex]\(-3 x^3\)[/tex] to each term inside the parentheses:
- Multiply [tex]\(-3 x^3\)[/tex] by [tex]\(x^2\)[/tex]:
[tex]\[
-3 x^3 \cdot x^2 = -3 x^{3+2} = -3 x^5
\][/tex]
- Multiply [tex]\(-3 x^3\)[/tex] by [tex]\(-2x\)[/tex]:
[tex]\[
-3 x^3 \cdot (-2x) = -3 \cdot -2 \cdot x^{3+1} = 6 x^4
\][/tex]
- Multiply [tex]\(-3 x^3\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[
-3 x^3 \cdot (-4) = -3 \cdot -4 \cdot x^3 = 12 x^3
\][/tex]
3. Combine the results from each distribution:
- From the first distribution, we have [tex]\(-3 x^5\)[/tex]
- From the second distribution, we have [tex]\(6 x^4\)[/tex]
- From the third distribution, we have [tex]\(12 x^3\)[/tex]
4. Combine the like terms to get the final simplified expression:
[tex]\[
-3 x^5 + 6 x^4 + 12 x^3
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
-3 x^5 + 6 x^4 + 12 x^3
\][/tex]
