If two polynomial expressions in $x$ are equal for every real number $x$, what must be true about the coefficients of $x^2$, $x$, and the constant terms on each side?

First, compare the constant terms on both sides to find $a$. Then, use the $x$-coefficients to find $b$. After you know both $a$ and $b$, compute their sum.

In Desmos, type y1 = (x - 3)(x + a) and y2 = x^2 + b*x - 21. When prompted, create sliders for a and b so you can adjust their values.

Move the sliders for a and b until the two parabolas overlap perfectly (you should see only one parabola because $y1$ and $y2$ are identical for all visible $x$). Note the values of a and b that make the graphs coincide.

Using the values of a and b you found, type an expression like a + b into Desmos (with those numbers substituted if needed) and read off the numerical result; that value is what the question is asking for.

Start by expanding $(x - 3)(x + a)$ using distribution (FOIL):

• Multiply $x$ by each term in $(x + a)$: $x \cdot x = x^2$ and $x \cdot a = ax$.
• Multiply $-3$ by each term in $(x + a)$: $-3 \cdot x = -3x$ and $-3 \cdot a = -3a$.
• Add all these terms together:

Combine like terms ($ax - 3x$ are both $x$-terms):

So the left-hand side becomes $x^2 + (a - 3)x - 3a$. The equation is now

Because these two quadratic expressions are equal for all real numbers $x$, the coefficients of the matching powers of $x$ must be the same on both sides.

Compare the constant terms (the terms without $x$):

• Left side constant: $-3a$
• Right side constant: $-21$

Set them equal:

Solve for $a$:

Now compare the coefficients of $x$:

• Left side $x$-coefficient: $a - 3$
• Right side $x$-coefficient: $b$

So

Substitute $a = 7$:

Finally, add $a$ and $b$:

So the value of $a + b$ is 11.