Question: k5 + 3k4 - 5k3 - 15k2 + 4k + 12 with every place with k place in a -1
1. **State the problem:**
Evaluate the expression $$k^5 + 3k^4 - 5k^3 - 15k^2 + 4k + 12$$ when $$k = -1$$.
2. **Substitute the value:**
Replace every $$k$$ with $$-1$$:
$$(-1)^5 + 3(-1)^4 - 5(-1)^3 - 15(-1)^2 + 4(-1) + 12$$
3. **Calculate each term:**
- $$(-1)^5 = -1$$
- $$3(-1)^4 = 3 \times 1 = 3$$
- $$-5(-1)^3 = -5 \times (-1) = 5$$
- $$-15(-1)^2 = -15 \times 1 = -15$$
- $$4(-1) = -4$$
- $$12$$ remains as $$12$$
4. **Sum all terms:**
$$-1 + 3 + 5 - 15 - 4 + 12$$
5. **Simplify step-by-step:**
$$(-1 + 3) = 2$$
$$2 + 5 = 7$$
$$7 - 15 = -8$$
$$-8 - 4 = -12$$
$$-12 + 12 = 0$$
6. **Final answer:**
$$0$$