Certainly! Let's solve the problem step by step to determine which expression is equivalent to \(92 x^2 + x - 1 + xy + \frac{2 x^3}{x}\).
First, let's start by simplifying the given expression:
1. The original expression is:
[tex]\[
92 x^2 + x - 1 + xy + \frac{2 x^3}{x}
\][/tex]
2. Simplify \(\frac{2 x^3}{x}\):
[tex]\[
\frac{2 x^3}{x} = 2 x^2
\][/tex]
3. Substitute this simplified term back into the original expression:
[tex]\[
92 x^2 + x - 1 + xy + 2 x^2
\][/tex]
4. Combine like terms \(92 x^2\) and \(2 x^2\):
[tex]\[
92 x^2 + 2 x^2 = 94 x^2
\][/tex]
Therefore, the simplified expression is:
[tex]\[
94 x^2 + x - 1 + xy
\][/tex]
Now, let's look at each provided option to see which one matches \(94 x^2 + x - 1 + xy\) exactly:
- Option A \(94 x^2 + x + xy - 1\):
[tex]\[
94 x^2 + x + xy - 1
\][/tex]
This matches our simplified expression exactly.
- Option B \(92 x^2 + 3 x + xy - 1\):
[tex]\[
92 x^2 + 3 x + xy - 1
\][/tex]
This does not match our simplified expression. The \(92 x^2\) term and the \(3 x\) term are not equivalent to \(94 x^2\) and \(x\).
- Option C \(2 x^2 + 92 x + y - 1\):
[tex]\[
2 x^2 + 92 x + y - 1
\][/tex]
This does not match our simplified expression. The terms are arranged differently, and their coefficients do not match \(94 x^2 + x - 1 + xy\).
- Option D \(2 x^3 + 92 x^2 + 2 x + xy - 1\):
[tex]\[
2 x^3 + 92 x^2 + 2 x + xy - 1
\][/tex]
This does not match our simplified expression as it contains a \(2 x^3\) term, which is not in our simplified expression.
Therefore, the expression that is equivalent to \(92 x^2 + x - 1 + xy + \frac{2 x^3}{x}\) is:
[tex]\[ \boxed{A} \][/tex]
