Replace $(x-2)(x+2)$ with $x^2-4$ to make the products shorter. (Anікo.аі)

Since the question asks for $a+b+c$, focus on the coefficients of $x^4$, $x^3$, and $x^2$ after you combine like terms.

In Desmos, define

• N=(3x-1)*(x^2-4)+5*(x+2)*(x^2+1)-(2x+7)*(x-2)*(x^2+1)

This is the numerator you get after multiplying by the common denominator.

Desmos will display N= as a polynomial in $x$. Make sure you can see the terms in descending powers (from $x^4$ down to the constant).

From the displayed polynomial, identify the coefficients of $x^4$, $x^3$, and $x^2$ (these are $a$, $b$, and $c$). © аnіko‌.аi

Add the three coefficients you found and select the answer choice that matches that sum.

Multiply both sides by $(x-2)(x+2)(x^2+1)$ to get a polynomial identity:

Note that $(x-2)(x+2)=x^2-4$.

Rewrite and expand each term:

anіk‌ο.аі

First expand $(x-2)(x^2+1)$:

Then multiply by $(2x+7)$:

So the last term is

Add the three expanded results:

• $x^4$ term: $-2x^4$ so $a=-2$
• $x^3$ term: $3x^3+5x^3-3x^3=5x^3$ so $b=5$
• $x^2$ term: $-x^2+10x^2+12x^2=21x^2$ so $c=21$

Therefore,