After clearing the denominator, you will get an equation of the form $x^3 + ax^2 + bx + c = \text{(something)}$. Work out what that "something" is by expanding $(x + 4)(x^2 + 3x + 2)$ and then adding 5.

Once you have both sides written as polynomials in standard form $x^3 + (\text{number})x^2 + (\text{number})x + (\text{number})$, compare the coefficients of $x$ on both sides to identify $b$.

In a new expression line, type (x+4)*(x^2+3x+2)+5 and press Enter. Desmos will display the expanded polynomial in standard form. The coefficient of x in this expanded polynomial is the value of $b$.

Start from the given equation:

Multiply both sides by the common denominator $x^2 + 3x + 2$ (which is nonzero in the domain) to remove the fractions:

Now expand $(x + 4)(x^2 + 3x + 2)$ term by term:

• $x \cdot x^2 = x^3$
• $x \cdot 3x = 3x^2$
• $x \cdot 2 = 2x$
• $4 \cdot x^2 = 4x^2$
• $4 \cdot 3x = 12x$
• $4 \cdot 2 = 8$

Combine like terms:

Now include the +5:

From Step 1, we have

and from Step 2, the right side simplifies to

So we must have

Matching coefficients of like powers of $x$:

• Coefficient of $x^2$: $a = 7$
• Coefficient of $x$: $b = 14$
• Constant term: $c = 13$

Therefore, the value of $b$ is $14$.