To resolve the expression $2 \cos^2 \theta + 13 \cos \theta + 15$ into factors. Step 1: Let $x = \cos \theta$. Substitute $x$ into the expression to form a quadratic equation. $$ 2x^2 + 13x + 15 $$ Step 2: Factor the quadratic expression by splitting the middle term. We need two numbers that multiply to $2 \times 15 = 30$ and add up to $13$. These numbers are $3$ and $10$. $$ 2x^2 + 3x + 10x + 15 $$ Step 3: Group the terms and factor out common factors from each group. $$ (2x^2 + 3x) + (10x + 15) $$ $$ x(2x + 3) + 5(2x + 3) $$ Step 4: Factor out the common binomial factor $(2x + 3)$. $$ (2x + 3)(x + 5) $$ Step 5: Substitute back $\cos \theta$ for $x$. $$ (2 \cos \theta + 3)(\cos \theta + 5) $$ The factored expression is: $$ \boxed{(2 \cos \theta + 3)(\cos \theta + 5)} $$