Let's solve the problem step-by-step.
Given:
[tex]\[ M(x) = 4x^2 - 3x \][/tex]
[tex]\[ N(x) = -5x^3 - 6x^2 - 3 \][/tex]
We need to find \( M(x) + N(x) \).
1. Identify the terms in \( M(x) \):
- \( 4x^2 \) (a term with \( x^2 \))
- \( -3x \) (a term with \( x \))
2. Identify the terms in \( N(x) \):
- \( -5x^3 \) (a term with \( x^3 \))
- \( -6x^2 \) (a term with \( x^2 \))
- \( -3 \) (a constant term)
3. Combine like terms:
- There are no \( x^3 \) terms in \( M(x) \), so the \( x^3 \) term in the sum is simply \( -5x^3 \).
- The \( x^2 \) terms are \( 4x^2 \) from \( M(x) \) and \( -6x^2 \) from \( N(x) \). Combined, they make:
[tex]\[ 4x^2 - 6x^2 = -2x^2 \][/tex]
- The \( x \) term from \( M(x) \) is \( -3x \), and there are no \( x \) terms in \( N(x) \). So the \( x \) term in the sum is \( -3x \).
- The constant term is \( -3 \) from \( N(x) \), and there are no constant terms in \( M(x) \). So the constant term in the sum is \( -3 \).
4. Combine all the terms:
- The combined expression is:
[tex]\[ -5x^3 - 2x^2 - 3x - 3 \][/tex]
Therefore, the sum \( M(x) + N(x) \) is:
[tex]\[ M(x) + N(x) = -5x^3 - 2x^2 - 3x - 3 \][/tex]
Thus, the correct answer is \( \boxed{-5x^3 - 2x^2 - 3x - 3} \).
Comparing this with the provided options, the correct choice is:
B. [tex]\( -5x^3 - 2x^2 - 3x - 3 \)[/tex]
