1. The problem is to analyze and understand the expression $\cos x (3 + 2 \cos x)$.\n\n2. This expression is a product of $\cos x$ and the binomial $(3 + 2 \cos x)$.\n\n3. We can expand the expression using the distributive property: $$\cos x (3 + 2 \cos x) = 3 \cos x + 2 \cos^2 x.$$\n\n4. This expression combines a linear term in $\cos x$ and a quadratic term in $\cos x$.\n\n5. To understand its behavior, note that $\cos x$ ranges between $-1$ and $1$.\n\n6. The minimum value occurs when $\cos x = -1$: $$3(-1) + 2(-1)^2 = -3 + 2 = -1.$$\n\n7. The maximum value occurs when $\cos x = 1$: $$3(1) + 2(1)^2 = 3 + 2 = 5.$$\n\n8. So, the expression ranges from $-1$ to $5$ as $x$ varies.\n\n9. This function can be graphed as $y = 3 \cos x + 2 \cos^2 x$.