To simplify the expression and classify the resulting polynomial, we start with the given expression:
[tex]\[
4x(x+1) - (3x - 8)(x + 4)
\][/tex]
### Step 1: Expand each term individually
First, expand [tex]\(4x(x + 1)\)[/tex]:
[tex]\[
4x(x + 1) = 4x^2 + 4x
\][/tex]
Next, expand [tex]\((3x - 8)(x + 4)\)[/tex]:
[tex]\[
(3x - 8)(x + 4) = 3x(x + 4) - 8(x + 4)
\][/tex]
Expanding each part separately:
[tex]\[
3x(x + 4) = 3x^2 + 12x
\][/tex]
[tex]\[
-8(x + 4) = -8x - 32
\][/tex]
Combining the expanded terms of [tex]\((3x - 8)(x + 4)\)[/tex]:
[tex]\[
3x^2 + 12x - 8x - 32 = 3x^2 + 4x - 32
\][/tex]
### Step 2: Combine all parts
Now we combine the expanded expressions:
[tex]\[
4x^2 + 4x - (3x^2 + 4x - 32)
\][/tex]
### Step 3: Distribute the subtraction
[tex]\[
4x^2 + 4x - 3x^2 - 4x + 32
\][/tex]
### Step 4: Combine like terms
Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
4x^2 - 3x^2 = x^2
\][/tex]
Combine the [tex]\(x\)[/tex] terms:
[tex]\[
4x - 4x = 0
\][/tex]
And then add the constant term:
[tex]\[
x^2 + 32
\][/tex]
The simplified expression is:
[tex]\[
x^2 + 32
\][/tex]
### Step 5: Classify the polynomial
Let's classify the polynomial [tex]\(x^2 + 32\)[/tex]:
- The degree of the polynomial is 2 because the highest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
- There are two terms in this polynomial: [tex]\(x^2\)[/tex] and 32.
A polynomial of degree two with two terms is known as a quadratic binomial. Therefore, the simplified polynomial, [tex]\(x^2 + 32\)[/tex], is:
### Answer
A. quadratic binomial
