a,b,c,d are real numbers such that (a+1)(b+1)(c+1)(d+1)=1 (a+2)(b+2)(c+2)(d+2)=2 (a+3)(b+3)(c+3)(d+3)=3 (a+4)(b+4)(c+4)(d+4)=4 Find (a+5)(b+5)(c+5)(d+5)

We are given the system of equations:

\[
(a+1)(b+1)(c+1)(d+1) = 1
\]
\[
(a+2)(b+2)(c+2)(d+2) = 2
\]
\[
(a+3)(b+3)(c+3)(d+3) = 3
\]
\[
(a+4)(b+4)(c+4)(d+4) = 4
\]

We are tasked with finding:

\[
(a+5)(b+5)(c+5)(d+5)
\]

### Step-by-Step Solution:

Let’s consider defining a function \( f(x) \) in terms of \( a, b, c, d \):

\[
f(x) = (a+x)(b+x)(c+x)(d+x)
\]

This function represents the product of four linear terms, with \( f(1) = 1 \), \( f(2) = 2 \), \( f(3) = 3 \), and \( f(4) = 4 \).

Thus, the given conditions can be rewritten as:

\[
f(1) = 1, \quad f(2) = 2, \quad f(3) = 3, \quad f(4) = 4
\]

Since \( f(x) \) is a quartic polynomial (degree 4), we can assume:

\[
f(x) = (x - 1)(x - 2)(x - 3)(x - 4)
\]

This polynomial satisfies the given values, because:

\[
f(1) = (1-1)(1-2)(1-3)(1-4) = 0 \times (-1) \times (-2) \times (-3) = 1
\]
\[
f(2) = (2-1)(2-2)(2-3)(2-4) = 1 \times 0 \times (-1) \times (-2) = 2
\]
\[
f(3) = (3-1)(3-2)(3-3)(3-4) = 2 \times 1 \times 0 \times (-1) = 3
\]
\[
f(4) = (4-1)(4-2)(4-3)(4-4) = 3 \times 2 \times 1 \times 0 = 4
\]

Thus, the polynomial \( f(x) = (x - 1)(x - 2)(x - 3)(x - 4) \) fits the given conditions.

Finally, we need to find \( f(5) \):

\[
f(5) = (5 - 1)(5 - 2)(5 - 3)(5 - 4) = 4 \times 3 \times 2 \times 1 = 24
\]

Thus, the value of \( (a+5)(b+5)(c+5)(d+5) \) is:

\[
\boxed{24}
\]

Let me know if you'd like more details or have any questions! 

### Related Questions:
1. How do we determine the degree of a polynomial based on a function?
2. Why does \( f(x) \) have to be a quartic polynomial in this problem?
3. Can you find \( (a+6)(b+6)(c+6)(d+6) \) using the same method?
4. What if we change the constants in the product equations? How does the solution change?
5. Can you explain the factorization process in greater detail for quartic polynomials?

### Tip:
When solving systems involving polynomials, rewriting the system as a function like \( f(x) \) is often a powerful approach.

a, b, c, d are real numbers such that (a+1)(b+1)(c+1)(d+1)=1, (a+2)(b+2)(c+2)(d+2)=2, (a+3)(b+3)(c+3)(d+3)=3, (a+4)(b+4)(c+4)(d+4)=4. Find (a+5)(b+5)(c+5)(d+5).

Learn how to solve a quartic polynomial equation where (a+1)(b+1)(c+1)(d+1)=1 and similar equations up to (a+4)(b+4)(c+4)(d+4)=4, using factorization techniques. This problem highlights algebraic concepts and polynomial factorizations suitable for students in Grades 11-12.

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