Certainly! Let's simplify the given algebraic expression step-by-step:
The expression is:
[tex]\[
(4x^3 + 3x^2 - 6x) - (10x^3 + 3x^2)
\][/tex]
### Step 1: Distribute the negative sign
First, we distribute the negative sign to each term inside the second parenthesis:
[tex]\[
(4x^3 + 3x^2 - 6x) - 10x^3 - 3x^2
\][/tex]
### Step 2: Combine like terms
Now, let's combine like terms by grouping the coefficients of the same powers of [tex]\( x \)[/tex]:
[tex]\[
4x^3 - 10x^3 + 3x^2 - 3x^2 - 6x
\][/tex]
Simplifying each group of like terms:
[tex]\[
(4 - 10)x^3 + (3 - 3)x^2 - 6x
\][/tex]
This reduces to:
[tex]\[
-6x^3 + 0x^2 - 6x
\][/tex]
Since [tex]\( 0x^2 \)[/tex] effectively cancels out, we are left with:
[tex]\[
-6x^3 - 6x
\][/tex]
### Conclusion
The simplified expression is:
[tex]\[
-6x^3 - 6x
\][/tex]
This simplified expression is a polynomial since it is a sum of terms consisting of a constant multiplied by a variable raised to a non-negative integer power.
### Choice Explanation
The correct choice corresponding to our simplification and confirming that it is a polynomial is:
[tex]\[
-6x^3 - 6x; \text{ is a polynomial}
\][/tex]
Thus, the correct answer is:
[tex]\[
-6x^3 - 6x; \text{ is a polynomial}
\][/tex]
