To use synthetic substitution, we evaluate the polynomial $f(x) = 2x^4 + 16x^3 - 28x^2 - 8x$ at $x = -2$. Here's the process:

Steps:

1. Write the coefficients of the polynomial:
   $2, 16, -28, -8, 0$
   (Note: Include the constant term and place $0$ for missing terms).

2. Set up synthetic division table:
   Use $-2$ as the divisor.

Table:

We perform the synthetic substitution calculation below:

Step	Coefficients $2, 16, -28, -8, 0$
Write $-2$ at left side
First number directly comes down: $2$
Multiply $2 \times -2 = -4$, add to $16$: 12
... Write # Perform synthetic substitution for f(x) = 2x^4 + 16x^3 - 28x^2 - 8x at x = -2

Coefficients of the polynomial

coefficients = [2, 16, -28, -8, 0] # Include 0 for the missing constant term x_value = -2

Perform synthetic substitution

synthetic_result = [] remainder = 0

for coeff in coefficients: remainder = coeff + remainder * x_value synthetic_result.append(remainder)

synthetic_result[-1] # The final remainder gives f(-2)-192-192successUsing synthetic substitution, the value of $f(-2)$ for $f(x) = 2x^4 + 16x^3 - 28x^2 - 8x$ is $\mathbf{-192}$.

Would you like details on how the calculation was performed or an explanation of synthetic substitution? Here are related questions:

1. How is synthetic substitution different from direct evaluation of a polynomial?
2. Can synthetic substitution be used for polynomials with missing terms?
3. How do you handle fractional or decimal roots in synthetic substitution?
4. What are the benefits of synthetic substitution for large polynomials?
5. How do you modify synthetic substitution for higher-order roots?

Tip: Always align the powers of $x$ in decreasing order to avoid errors in synthetic substitution.