When two polynomials are equal for all real $x$, the coefficients of $x^3$, $x^2$, $x$, and the constant term on both sides must match. Write equations by matching each coefficient from the left-hand side to the right-hand side.

Solve the coefficient equations in a sensible order: first find $a$, then $b$, then $c$. After you have all three, remember the question asks for $a + b + c$, not for $a$, $b$, and $c$ separately.

In Desmos, type y1 = (x^2 + a*x + b)*(x + 4) + c and accept the sliders for $a$, $b$, and $c$. On a new line, type y2 = x^3 + 9x^2 + 4x - 6 so both expressions are graphed.

Move the sliders for $a$, $b$, and $c$ until the graphs of $y1$ and $y2$ overlap everywhere (they should appear as a single curve). You can zoom in or use a table of values to confirm that $y1$ and $y2$ match at several $x$-values.

Once the sliders give you exact integer values for $a$, $b$, and $c$, add a new line and type a + b + c. Desmos will display the numerical value of this expression, which is the required answer.

Start by expanding $(x^2 + ax + b)(x + 4)$.

Now include the $+c$ term:

So the left-hand side in standard form is $x^3 + (4+a)x^2 + (4a+b)x + 4b + c$. The right-hand side is already in standard form: $x^3 + 9x^2 + 4x - 6$.

Because the two polynomials are equal for all real $x$, the coefficients of corresponding powers of $x$ must match.

From $x^3$ terms:

• Left: $1$
• Right: $1$

This is already satisfied.

From $x^2$ terms:

From $x$ terms:

From constant terms:

Now solve these equations one by one.

From $4 + a = 9$:

Substitute $a = 5$ into $4a + b = 4$:

Now substitute $b = -16$ into $4b + c = -6$:

So $a = 5$, $b = -16$, and $c = 58$.

The question asks for $a + b + c$, not the individual values.

Using $a = 5$, $b = -16$, and $c = 58$:

First combine $5$ and $-16$:

Then add $58$:

So the value of $a + b + c$ is 47.