What is the sum of a, b, and c when 2x^4 - x^3 + 4 is divided by x + 1 yielding the expression 2x^3 - 3x^2 + ax + b + x + 1?

Steps to Solve

1. Set Up the Polynomial Division Identify the polynomial that you are dividing and the divisor. For example, if you have a division of the form:

   $$ P(x) = (x^3 + ax^2 + bx + c) \div (x - r) $$

   You would set this up to find the quotient and remainder, if necessary.

2. Identify the Expression Resulting from Division If the question gives a resulting polynomial or expression after the division, set that equal to the polynomial you created in step 1. For example, set:

   $$ \text{Result} = Q(x) + R $$

   Where $Q(x)$ is the quotient, and $R$ is the remainder.

3. Combine Like Terms By equating the two sides of the polynomial, combine like terms to create equations for the coefficients of $x^2$, $x$, and the constant term. For example, you might get:

   $$ a = \text{coefficient from Result of division} $$

   $$ b = \text{next coefficient from Result of division} $$

   $$ c = \text{constant term from Result of division} $$

4. Solve for a, b, and c From the equations formed in the previous step, solve for the variables $a$, $b$, and $c$. This could involve substitution or simply solving linear equations.

5. Calculate the Sum a + b + c Finally, add the values of $a$, $b$, and $c$ together to get the final result:

   $$ S = a + b + c $$