To solve the problem of simplifying the expression [tex]\( (x - 1)(x + 2)(x + 3)(x + 4) + 6 \)[/tex], let's go through it step by step.
### Step 1: Expand [tex]\( (x - 1)(x + 2) \)[/tex] and [tex]\( (x + 3)(x + 4) \)[/tex]
First, we'll simplify the inner products:
[tex]\[ (x - 1)(x + 2) \][/tex]
[tex]\[ = x(x + 2) - 1(x + 2) \][/tex]
[tex]\[ = x^2 + 2x - x - 2 \][/tex]
[tex]\[ = x^2 + x - 2 \][/tex]
Similarly, for the second product:
[tex]\[ (x + 3)(x + 4) \][/tex]
[tex]\[ = x(x + 4) + 3(x + 4) \][/tex]
[tex]\[ = x^2 + 4x + 3x + 12 \][/tex]
[tex]\[ = x^2 + 7x + 12 \][/tex]
### Step 2: Multiply the two expanded forms
Next, we'll multiply the results from step 1:
[tex]\[ (x^2 + x -2)(x^2 + 7x + 12) \][/tex]
We need to distribute each term in the first polynomial to each term in the second polynomial:
[tex]\[ = (x^2 + x - 2)(x^2) + (x^2 + x - 2)(7x) + (x^2 + x - 2)(12) \][/tex]
Distribute each term:
[tex]\[ = x^2 \cdot x^2 + x^2 \cdot 7x + x^2 \cdot 12 + x \cdot x^2 + x \cdot 7x + x \cdot 12 - 2 \cdot x^2 - 2 \cdot 7x - 2 \cdot 12 \][/tex]
[tex]\[ = x^4 + 7x^3 + 12x^2 + x^3 + 7x^2 + 12x - 2x^2 - 14x - 24 \][/tex]
Combine like terms:
[tex]\[ = x^4 + (7x^3 + x^3) + (12x^2 + 7x^2 - 2x^2) + (12x - 14x) - 24 \][/tex]
[tex]\[ = x^4 + 8x^3 + 17x^2 - 2x - 24 \][/tex]
### Step 3: Add the constant 6
Now we add the constant term 6 to the expanded polynomial:
[tex]\[ x^4 + 8x^3 + 17x^2 - 2x - 24 + 6 \][/tex]
Combine the constant terms:
[tex]\[ = x^4 + 8x^3 + 17x^2 - 2x - 18 \][/tex]
### Conclusion
The simplified form of the expression [tex]\( (x - 1)(x + 2)(x + 3)(x + 4) + 6 \)[/tex] is:
[tex]\[ x^4 + 8x^3 + 17x^2 - 2x - 18 \][/tex]
Thus, the final answer is:
[tex]\[ (x - 1)(x + 2)(x + 3)(x + 4) + 6 = x^4 + 8x^3 + 17x^2 - 2x - 18 \][/tex]
