Question 141·Medium·Advanced Math

The problem excludes $-\frac34$, not $0$, so $0$ is safe to test. Type $x_1=0$, then type the given fraction with $x_1$ in place of $x$. Here $x_1$ names the input you're testing. Desmos prints $-1$, so an equivalent choice must also give $-1$ at $0$.

Type each choice on a new Desmos line, replacing $x$ with $x_1$. Desmos prints $-0.25$, $-0.25$, $1$, and $-1$, in order. Only the last choice matches the original at $0$. A match at one input isn't proof on its own, but every equivalent choice must match there, so the other three are out. The equivalent expression is $3x-1$ for the allowed inputs $x\ne-\frac34$. Choice D.

No Desmos needed. Factoring explains the match for every allowed input. To factor $12x^2+5x-3$, split the middle term using two numbers that add to $5$ and multiply to $12(-3)=-36$. Those numbers are $9$ and $-4$.

Split $5x$ into $9x-4x$:

Factor each pair:

Pull out the shared factor $4x+3$:

Replace the numerator with its factored form:

Cancel the whole factor $4x+3$, which is nonzero because the problem says $x\ne-\frac34$:

So the equivalent expression is $3x-1$ on the original fraction's allowed inputs. Choice D.