After clearing the denominator, expand $(2x^2 + ax + b)(x^2 - 2x + 2)$ by distributing each term in the first factor across the second factor, then combine like terms.

Once both sides are written as standard polynomials in $x$, compare the coefficients of each power of $x$ on both sides. Focus on the $x^3$ terms to find $a$ without needing to solve for $b$, $c$, and $d$.

In Desmos, type

f(x) = (2x^4 - 3x^3 + 7x^2 - 12x + 8) / (x^2 - 2x + 2)

This is the given expression you are rewriting.

After you solve the problem on paper and find numerical values for $a$, $b$, $c$, and $d$, type in Desmos:

g(x) = 2x^2 + (your a)*x + (your b) + ((your c)*x + (your d)) / (x^2 - 2x + 2)

Replace your a, your b, your c, and your d with the numbers you found.

Look at the graphs of $f(x)$ and $g(x)$. If they lie exactly on top of each other for all visible $x$-values, then your values of $a$, $b$, $c$, and $d$ are consistent with the original expression, and the coefficient you found for $a$ is correct.

Start from the given identity:

Multiply both sides by $x^2 - 2x + 2$ to remove the fractions:

Now you have an equality between polynomials that must hold for every value of $x$. This means the coefficients (the numbers in front of each power of $x$) on both sides must match.

Expand $(2x^2 + ax + b)(x^2 - 2x + 2)$ term by term:

• $2x^2(x^2 - 2x + 2) = 2x^4 - 4x^3 + 4x^2$
• $ax(x^2 - 2x + 2) = a x^3 - 2a x^2 + 2a x$
• $b(x^2 - 2x + 2) = b x^2 - 2b x + 2b$

Add these together:

• $x^4$ term: $2x^4$
• $x^3$ term: $(-4 + a)x^3$
• $x^2$ term: $(4 - 2a + b)x^2$
• $x$ term: $(2a - 2b)x$
• Constant term: $2b$

Now include the remainder $(cx + d)$:

The entire right-hand side becomes

Now compare this expanded right-hand side to the left-hand side polynomial

Match coefficients of each power of $x$.

For the $x^3$ terms:

• Left side coefficient of $x^3$ is $-3$.
• Right side coefficient of $x^3$ is $-4 + a$.

Set them equal:

Solve for $a$:

So the value of $a$ is $1$, which corresponds to answer choice C.