Answer :
To simplify the expression \( 4x(x + 1) - (3x - 8)(x + 4) \), let's break it down step-by-step.
### Step 1: Expand the first part \( 4x(x + 1) \)
Expanding this expression:
[tex]\[ 4x(x + 1) = 4x \cdot x + 4x \cdot 1 = 4x^2 + 4x \][/tex]
### Step 2: Expand the second part \( (3x - 8)(x + 4) \)
Expanding this expression:
[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]
### Step 3: Subtract the second expanded expression from the first expanded expression
Now, we subtract the expanded form of the second expression from the expanded form of the first expression:
[tex]\[ 4x^2 + 4x - (3x^2 + 4x - 32) \][/tex]
[tex]\[ = 4x^2 + 4x - 3x^2 - 4x + 32 \][/tex]
### Step 4: Simplify the resulting expression
Combining like terms:
[tex]\[ = (4x^2 - 3x^2) + (4x - 4x) + 32 \][/tex]
[tex]\[ = x^2 + 0 + 32 \][/tex]
[tex]\[ = x^2 + 32 \][/tex]
### Step 5: Classify the resulting polynomial
We observe that the simplified expression is \( x^2 + 32 \). This is a quadratic polynomial because the highest degree of \( x \) is 2. The expression has two terms: \( x^2 \) and \( 32 \). Therefore, it is a binomial (a polynomial containing exactly two terms).
Thus, the resulting polynomial is a quadratic binomial.
The answer is:
B. quadratic binomial
### Step 1: Expand the first part \( 4x(x + 1) \)
Expanding this expression:
[tex]\[ 4x(x + 1) = 4x \cdot x + 4x \cdot 1 = 4x^2 + 4x \][/tex]
### Step 2: Expand the second part \( (3x - 8)(x + 4) \)
Expanding this expression:
[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]
### Step 3: Subtract the second expanded expression from the first expanded expression
Now, we subtract the expanded form of the second expression from the expanded form of the first expression:
[tex]\[ 4x^2 + 4x - (3x^2 + 4x - 32) \][/tex]
[tex]\[ = 4x^2 + 4x - 3x^2 - 4x + 32 \][/tex]
### Step 4: Simplify the resulting expression
Combining like terms:
[tex]\[ = (4x^2 - 3x^2) + (4x - 4x) + 32 \][/tex]
[tex]\[ = x^2 + 0 + 32 \][/tex]
[tex]\[ = x^2 + 32 \][/tex]
### Step 5: Classify the resulting polynomial
We observe that the simplified expression is \( x^2 + 32 \). This is a quadratic polynomial because the highest degree of \( x \) is 2. The expression has two terms: \( x^2 \) and \( 32 \). Therefore, it is a binomial (a polynomial containing exactly two terms).
Thus, the resulting polynomial is a quadratic binomial.
The answer is:
B. quadratic binomial
