Answer :
To subtract the polynomial
$$5x^3 + 4x^2$$
by
$$6x^2 - 2x - 9,$$
we begin by writing the expression:
$$
(5x^3 + 4x^2) - (6x^2 - 2x - 9).
$$
**Step 1. Distribute the minus sign:**
We multiply each term inside the second set of parentheses by $-1$:
$$
5x^3 + 4x^2 - 6x^2 + 2x + 9.
$$
**Step 2. Combine like terms:**
- The $x^3$ term: There is only one term: $5x^3$.
- The $x^2$ terms: $4x^2 - 6x^2$ gives $-2x^2$.
- The $x$ term: There is only one term: $2x$.
- The constant term: There is only one term: $9$.
Thus, combining the like terms, we obtain:
$$
5x^3 - 2x^2 + 2x + 9.
$$
**Final Answer:**
The difference of the given polynomials is
$$
\boxed{5x^3 - 2x^2 + 2x + 9}.
$$
$$5x^3 + 4x^2$$
by
$$6x^2 - 2x - 9,$$
we begin by writing the expression:
$$
(5x^3 + 4x^2) - (6x^2 - 2x - 9).
$$
**Step 1. Distribute the minus sign:**
We multiply each term inside the second set of parentheses by $-1$:
$$
5x^3 + 4x^2 - 6x^2 + 2x + 9.
$$
**Step 2. Combine like terms:**
- The $x^3$ term: There is only one term: $5x^3$.
- The $x^2$ terms: $4x^2 - 6x^2$ gives $-2x^2$.
- The $x$ term: There is only one term: $2x$.
- The constant term: There is only one term: $9$.
Thus, combining the like terms, we obtain:
$$
5x^3 - 2x^2 + 2x + 9.
$$
**Final Answer:**
The difference of the given polynomials is
$$
\boxed{5x^3 - 2x^2 + 2x + 9}.
$$
