Question 60·Medium·Advanced Math

To use the given difference, put each function's formula in its place:

Keep parentheses around $q(x)$ because you're subtracting the whole polynomial.

Follow the squared terms because $a$ is the coefficient, or number multiplying $x^2$, in $p(x)$. Subtract $q$'s squared term:

That $x^2$ is subtracted, not added.

The difference equals the given expression for every $x$, so terms with the same power must have the same coefficient. Match the coefficients of $x^2$:

To solve and check the full identity in Desmos, type $x_{1}=[1...5]$, then $(ax_{1}^{2}+3x_{1}+2)-(x_{1}^{2}-5x_{1}+4)\sim4x_{1}^{2}+8x_{1}-2$. The list supplies several inputs, and $\sim$ tells Desmos to fit $a$. Under PARAMETERS, Desmos reports $a=5$. So the coefficient in $p(x)$ is $5$. Choice C.