To solve and classify the given expression [tex]\( 4x(x+1) - (3x-8)(x+4) \)[/tex]:
1. Distribute and expand each part of the expression:
- [tex]\( 4x(x+1) \)[/tex]:
[tex]\[
4x \cdot x + 4x \cdot 1 = 4x^2 + 4x
\][/tex]
- [tex]\((3x-8)(x+4)\)[/tex]:
[tex]\[
(3x-8)(x+4) = 3x \cdot x + 3x \cdot 4 - 8 \cdot x - 8 \cdot 4 = 3x^2 + 12x - 8x - 32 = 3x^2 + 4x - 32
\][/tex]
2. Combine the two expressions:
The given expression is
[tex]\[
4x(x+1) - (3x-8)(x+4)
\][/tex]
Substituting the expanded forms, we get
[tex]\[
4x^2 + 4x - (3x^2 + 4x - 32)
\][/tex]
3. Distribute the minus sign through the second part:
[tex]\[
4x^2 + 4x - 3x^2 - 4x + 32
\][/tex]
4. Combine like terms:
[tex]\[
4x^2 - 3x^2 + 4x - 4x + 32
\][/tex]
[tex]\[
(4x^2 - 3x^2) + (4x - 4x) + 32
\][/tex]
[tex]\[
x^2 + 0 + 32
\][/tex]
Simplifies to:
[tex]\[
x^2 + 32
\][/tex]
5. Classify the resulting polynomial:
- The polynomial is [tex]\( x^2 + 32 \)[/tex].
- It has a degree of 2 (highest power of [tex]\( x \)[/tex] is 2).
- It has 2 distinct terms: [tex]\( x^2 \)[/tex] and 32.
Therefore, the simplified expression is [tex]\( x^2 + 32 \)[/tex] and it is a quadratic binomial.
So, the correct answer is:
A. quadratic binomial
