Sure, let's solve the expression step-by-step and classify the resulting polynomial.
We start with the given expression:
[tex]\[ 4x(x + 1) - (3x - 8)(x + 4) \][/tex]
First, let's expand [tex]\( 4x(x + 1) \)[/tex]:
[tex]\[ 4x(x + 1) = 4x^2 + 4x \][/tex]
Next, we expand [tex]\( (3x - 8)(x + 4) \)[/tex]:
[tex]\[ (3x - 8)(x + 4) \][/tex]
Use the distributive property (also known as the FOIL method for binomials):
[tex]\[ = 3x(x) + 3x(4) - 8(x) - 8(4) \][/tex]
[tex]\[ = 3x^2 + 12x - 8x - 32 \][/tex]
Combine like terms:
[tex]\[ = 3x^2 + 4x - 32 \][/tex]
Now, we have:
[tex]\[ 4x^2 + 4x - (3x^2 + 4x - 32) \][/tex]
Distribute the negative sign through the second polynomial:
[tex]\[ 4x^2 + 4x - 3x^2 - 4x + 32 \][/tex]
Combine like terms:
[tex]\[ (4x^2 - 3x^2) + (4x - 4x) + 32 \][/tex]
[tex]\[ = x^2 + 32 \][/tex]
The resulting expression is:
[tex]\[ x^2 + 32 \][/tex]
This is a quadratic polynomial because the highest power of [tex]\( x \)[/tex] is 2, and it has two terms, making it a binomial.
Therefore, the correct classification of the polynomial is:
A. quadratic binomial
