After expanding, you should have something like $2x^2 + (\text{something})x + (\text{something})$. Line this up with $2x^2 + 11x + 12$ and compare the coefficients.

Set the coefficient of $x$ from your expanded expression equal to $11$, and set the constant term equal to $12$. Solve for $b$ using one of these equations and check it with the other.

In Desmos, type y = (x + 4)(2x + b). When you press enter, Desmos will prompt you to add a slider for b; create that slider.

On the next line, type y = 2x^2 + 11x + 12 to graph the given expression.

Move the b slider until the graph of y = (x + 4)(2x + b) lies exactly on top of the graph of y = 2x^2 + 11x + 12 for all visible $x$-values. The value shown for b on the slider at that point is the solution.

Start by expanding the expression $(x + 4)(2x + b)$.

Combine like terms:

So $(x + 4)(2x + b)$ simplifies to $2x^2 + (b + 8)x + 4b$.

We are told that $(x + 4)(2x + b)$ is equivalent to $2x^2 + 11x + 12$.

From Step 1, we know:

Set this equal to the given expression:

Because the expressions are equivalent for all $x$, the coefficients of matching powers of $x$ must be the same on both sides.

Match the coefficient of $x$ and the constant term:

• Coefficient of $x$:

• Constant term:

Each of these equations can be used to solve for $b$. Use either one (and later verify with the other).

From $4b = 12$:

Check with the $x$-coefficient equation: $b + 8 = 3 + 8 = 11$, which matches the given expression.

So the value of $b$ is $3$, which corresponds to answer choice C.