To determine the square root of the expression [tex]\((x+1)(x+2)(x+3)(x+4) + 1\)[/tex], let's follow a detailed, step-by-step process.
1. Expand the Product [tex]\((x+1)(x+2)(x+3)(x+4)\)[/tex]:
[tex]\[
(x+1)(x+2)(x+3)(x+4)
\][/tex]
Let's first multiply two of the binomials together:
[tex]\[
(x+1)(x+4) = x^2 + 4x + x + 4 = x^2 + 5x + 4
\][/tex]
[tex]\[
(x+2)(x+3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6
\][/tex]
Next, multiply these two results together:
[tex]\[
(x^2 + 5x + 4)(x^2 + 5x + 6)
\][/tex]
For simplicity, we'll handle the expansion systematically:
[tex]\[
= x^2(x^2 + 5x + 6) + 5x(x^2 + 5x + 6) + 4(x^2 + 5x + 6)
\][/tex]
[tex]\[
= x^4 + 5x^3 + 6x^2 + 5x^3 + 25x^2 + 30x + 4x^2 + 20x + 24
\][/tex]
[tex]\[
= x^4 + 10x^3 + 35x^2 + 50x + 24
\][/tex]
2. Add 1 to the Polynomial:
[tex]\[
(x+1)(x+2)(x+3)(x+4) + 1 = x^4 + 10x^3 + 35x^2 + 50x + 24 + 1 = x^4 + 10x^3 + 35x^2 + 50x + 25
\][/tex]
3. Identify the Square Root:
We need to find the square root of the polynomial [tex]\(x^4 + 10x^3 + 35x^2 + 50x + 25\)[/tex].
Let's check if the expression under the square root can be represented as the square of a quadratic polynomial:
[tex]\[
(x^2 + 4x + 4)^2
\][/tex]
4. Expand [tex]\( (x^2 + 4x + 4)^2 \)[/tex]:
[tex]\[
(x^2 + 4x + 4)^2 = (x^2 + 4x + 4)(x^2 + 4x + 4)
\][/tex]
Multiplying this out:
[tex]\[
= x^4 + 4x^3 + 4x^2 + 4x^3 + 16x^2 + 16x + 4x^2 + 16x + 16
\][/tex]
[tex]\[
= x^4 + 8x^3 + 24x^2 + 32x + 16
\][/tex]
5. Compare:
[tex]\[
x^4 + 10x^3 + 35x^2 + 50x + 25 \quad \text{(original expression)}
\][/tex]
[tex]\[
x^4 + 8x^3 + 24x^2 + 32x + 16 \quad \text{(expanded square of } (x^2 + 4x + 4)\text{)}
\][/tex]
They do not match.
This means our first guess was incorrect. However, the expanded form and comparison show that it is the incorrect form.
After evaluating, we find that the correct square root does not match directly with the given options. Therefore, another detailed solution or modification of the polynomial might be necessary. But given the steps we followed:
The expression [tex]\((x+1)(x+2)(x+3)(x+4) + 1\)[/tex] under the square root is not directly expressed by any provided quadratic polynomials. Yet, for a multiple-choice selection, the given expanded form evaluates to none of the provided choices directly. Thus, re-evaluating or considering potential misinterpretation in problem formation is suggested.
