Sure, let's work through the given expression step by step to simplify it.
We start with the given expression:
[tex]\[ B = (x-3)^2 - (x-1)^2 + 4x \][/tex]
First, let's expand both squared terms:
1. Expanding [tex]\((x-3)^2\)[/tex]:
[tex]\[ (x-3)^2 = x^2 - 6x + 9 \][/tex]
2. Expanding [tex]\((x-1)^2\)[/tex]:
[tex]\[ (x-1)^2 = x^2 - 2x + 1 \][/tex]
Now substituting these expanded forms back into the original expression, we have:
[tex]\[ B = (x^2 - 6x + 9) - (x^2 - 2x + 1) + 4x \][/tex]
Next, distribute the negative sign across the second quadratic expression:
[tex]\[ B = x^2 - 6x + 9 - x^2 + 2x - 1 + 4x \][/tex]
Combine the like terms by adding or subtracting the coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant terms:
- The [tex]\(x^2\)[/tex] terms: [tex]\(x^2 - x^2 = 0\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-6x + 2x + 4x = 0\)[/tex]
- The constant terms: [tex]\(9 - 1 = 8\)[/tex]
Therefore, the expression simplifies to:
[tex]\[ B = 0 + 0 + 8 \][/tex]
Thus,
[tex]\[ B = 8 \][/tex]
So, the correct answer is:
e) 8
