Sure, let's break down the problem step by step:
1. Given Expression:
[tex]\[ 4x(x + 1) - (3x - 8)(x + 4) \][/tex]
2. Expanding Each Term Separately:
- First, let's expand [tex]\(4x(x + 1)\)[/tex]:
[tex]\[ 4x(x + 1) = 4x \cdot x + 4x \cdot 1 = 4x^2 + 4x \][/tex]
- Next, let's expand [tex]\((3x - 8)(x + 4)\)[/tex]:
Using the distributive property (FOIL method):
[tex]\[ (3x - 8)(x + 4) = 3x \cdot x + 3x \cdot 4 - 8 \cdot x - 8 \cdot 4 \][/tex]
[tex]\[ = 3x^2 + 12x - 8x - 32 \][/tex]
[tex]\[ = 3x^2 + 4x - 32 \][/tex]
3. Combine the Expanded Terms:
[tex]\[ 4x(x + 1) - (3x - 8)(x + 4) = (4x^2 + 4x) - (3x^2 + 4x - 32) \][/tex]
Distribute the negative sign:
[tex]\[ = 4x^2 + 4x - 3x^2 - 4x + 32 \][/tex]
4. Combine Like Terms:
[tex]\[ = (4x^2 - 3x^2) + (4x - 4x) + 32 \][/tex]
[tex]\[ = x^2 + 32 \][/tex]
5. Simplified Expression:
[tex]\[ x^2 + 32 \][/tex]
6. Classification:
- The polynomial is of degree 2 (the highest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex]).
- It has two terms: [tex]\(x^2\)[/tex] and [tex]\(32\)[/tex].
Thus, the simplified expression is [tex]\(x^2 + 32\)[/tex], which is a quadratic binomial (a binomial is a polynomial with exactly two terms).
Therefore, the correct answer is:
A. quadratic binomial
