Express $$ f(x) $$ in the form $$ f(x)=(x-k) q(x)+r $$ for the given value of $$ k $$. $$ f(x)=5 x^{3}+x^{2}+x-4, k=-1 $$

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To express $$ f(x) = 5x^3 + x^2 + x - 4 $$ in the form $$ f(x) = (x - k)q(x) + r $$ for $$ k = -1 $$, we will perform polynomial long division of $$ f(x) $$ by $$ x - (-1) = x + 1 $$.

1. **Set up the division**: We want to divide $$ f(x) $$ by $$ x + 1 $$.

2. **Perform the division**:
   - Divide the leading term of $$ f(x) $$ (which is $$ 5x^3 $$) by the leading term of $$ x + 1 $$ (which is $$ x $$):
     $$
     \frac{5x^3}{x} = 5x^2
     $$
   - Multiply $$ 5x^2 $$ by $$ x + 1 $$:
     $$
     5x^2(x + 1) = 5x^3 + 5x^2
     $$
   - Subtract this from $$ f(x) $$:
     $$
     (5x^3 + x^2 + x - 4) - (5x^3 + 5x^2) = (x^2 - 5x^2) + x - 4 = -4x^2 + x - 4
     $$

3. **Repeat the process**:
   - Divide the leading term $$ -4x^2 $$ by $$ x $$:
     $$
     \frac{-4x^2}{x} = -4x
     $$
   - Multiply $$ -4x $$ by $$ x + 1 $$:
     $$
     -4x(x + 1) = -4x^2 - 4x
     $$
   - Subtract this from the current remainder:
     $$
     (-4x^2 + x - 4) - (-4x^2 - 4x) = (x + 4x) - 4 = 5x - 4
     $$

4. **Repeat again**:
   - Divide the leading term $$ 5x $$ by $$ x $$:
     $$
     \frac{5x}{x} = 5
     $$
   - Multiply $$ 5 $$ by $$ x + 1 $$:
     $$
     5(x + 1) = 5x + 5
     $$
   - Subtract this from the current remainder:
     $$
     (5x - 4) - (5x + 5) = -4 - 5 = -9
     $$

5. **Final result**:
   - The division gives us:
     $$
     f(x) = (x + 1)(5x^2 - 4x + 5) - 9
     $$

Thus, we can express $$ f(x) $$ in the desired form:
$$
f(x) = (x + 1)(5x^2 - 4x + 5) - 9
$$

In conclusion, the expression for $$ f(x) $$ is:
$$
f(x) = (x + 1)q(x) + r \quad \text{where } q(x) = 5x^2 - 4x + 5 \text{ and } r = -9.
$$
$$ f(x) = (x + 1)(5x^2 - 4x + 5) - 9 $$

Express in the form for the given value of .

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