If `x^2+ax+1` is a factor of `ax^3+ bx + c`, then which of the following conditions are not valid: a. `a^2+c=0` b. `b-a=ac` c. `c^3+c+b^2=0` d. `2c+a=b`

Explanation

To solve the problem, we need to determine the conditions under which the polynomial \( x^2 + ax + 1 \) is a factor of \( ax^3 + bx + c \). We will do this by performing polynomial long division and analyzing the remainder.

### Step 1: Polynomial Division
We start by dividing \( ax^3 + bx + c \) by \( x^2 + ax + 1 \).

1. **First Division Step**:
- To get \( ax^3 \), we multiply \( x^2 + ax + 1 \) by \( ax \):
\[
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