Question 144·Hard·Advanced Math

Choose $x=1$: it doesn't make the denominator $x+3$ zero, and it leaves $kx^2$ equal to $k$. At $x=0$, the unknown would disappear. Since the expressions must agree at every allowed input, substitute $1$ on both sides:

One allowed input can determine $k$ because equivalence is given; one matching input alone cannot prove an identity.

Type the equation with $\sim$ instead of $=$ so Desmos finds the unknown $k$:

Desmos reports $k=12$ under PARAMETERS. The given equivalence must work at $x=1$, so the required integer is $12$. Choice C.

For every $x\ne-3$, the denominator $x+3$ is nonzero, so multiply both sides of the given equivalence by it:

The two sides must now be the same polynomial, with matching terms.

Distribute both terms of $x+3$ across $4x^2-9$:

The $x^2$ term comes from $3(4x^2)$, so its coefficient is $12$.

A coefficient is the number multiplying a power of $x$. The original numerator has $kx^2$; the product has $12x^2$, and their other terms already match. So the $x^2$ coefficients must agree:

This confirms the fraction equals the required expression for every allowed $x$. Choice C.