We are given that $f(x)$ is a fourth-degree polynomial of the form:

$f(x) = x^4 + ax^3 + bx^2 + cx + d$ where the leading coefficient of $x^4$ is 1. We are also provided with the following conditions:

$f(-1) = -1, \quad f(2) = -4, \quad f(-3) = -9, \quad f(4) = -16$

Our goal is to find the value of $f(1)$.

Step 1: Set up the equations

Using the conditions $f(-1) = -1$, $f(2) = -4$, $f(-3) = -9$, and $f(4) = -16$, we can substitute these values into the polynomial to create a system of equations.

1. For $f(-1) = -1$:

$(-1)^4 + a(-1)^3 + b(-1)^2 + c(-1) + d = -1$ $1 - a + b - c + d = -1 \quad \Rightarrow \quad -a + b - c + d = -2 \quad \text{(Equation 1)}$

2. For $f(2) = -4$:

$(2)^4 + a(2)^3 + b(2)^2 + c(2) + d = -4$ $16 + 8a + 4b + 2c + d = -4 \quad \Rightarrow \quad 8a + 4b + 2c + d = -20 \quad \text{(Equation 2)}$

3. For $f(-3) = -9$:

$(-3)^4 + a(-3)^3 + b(-3)^2 + c(-3) + d = -9$ $81 - 27a + 9b - 3c + d = -9 \quad \Rightarrow \quad -27a + 9b - 3c + d = -90 \quad \text{(Equation 3)}$

4. For $f(4) = -16$:

$(4)^4 + a(4)^3 + b(4)^2 + c(4) + d = -16$ $256 + 64a + 16b + 4c + d = -16 \quad \Rightarrow \quad 64a + 16b + 4c + d = -272 \quad \text{(Equation 4)}$

Step 2: Solve the system of equations

We now solve the system of four equations:

1. $-a + b - c + d = -2$
2. $8a + 4b + 2c + d = -20$
3. $-27a + 9b - 3c + d = -90$
4. $64a + 16b + 4c + d = -272$

Let me solve this system.The solution to the system of equations is:

$a = -2, \quad b = -14, \quad c = 14, \quad d = 24$

Thus, the polynomial $f(x)$ is:

$f(x) = x^4 - 2x^3 - 14x^2 + 14x + 24$

Step 3: Find $f(1)$

Now, we substitute $x = 1$ into the polynomial:

$f(1) = (1)^4 - 2(1)^3 - 14(1)^2 + 14(1) + 24$ $f(1) = 1 - 2 - 14 + 14 + 24 = 23$

Thus, the value of $f(1)$ is:

$f(1) = 23$

Let me know if you would like further details on the process.

Here are 5 related questions to expand your understanding:

1. How do you find the roots of a polynomial equation?
2. Can you explain the process of solving a system of nonlinear equations?
3. How do the degrees of polynomials affect their graph shapes?
4. What are the methods to factor a fourth-degree polynomial?
5. How can you determine if a polynomial has any symmetry properties?

Tip: Always check for special cases like symmetry or known root values when solving polynomial problems. It may simplify the process.