1. **State the problem:** Find the ratio $a : b$ given the identity $$x(2x + b) + a(x + 1) \equiv 2x^2 + 8cx + 2c$$ where $a$, $b$, and $c$ are non-zero constants. 2. **Write the expression on the left side:** $$x(2x + b) + a(x + 1) = 2x^2 + bx + ax + a$$ 3. **Group like terms:** $$2x^2 + (b + a)x + a$$ 4. **Equate coefficients with the right side:** Given $$2x^2 + (b + a)x + a \equiv 2x^2 + 8cx + 2c$$ we match coefficients of powers of $x$: - Coefficient of $x^2$: $2 = 2$ (already equal) - Coefficient of $x$: $b + a = 8c$ - Constant term: $a = 2c$ 5. **Substitute $a = 2c$ into $b + a = 8c$:** $$b + 2c = 8c$$ $$b = 8c - 2c = 6c$$ 6. **Express $b$ in terms of $a$:** Since $a = 2c$, then $c = \frac{a}{2}$, so $$b = 6c = 6 \times \frac{a}{2} = 3a$$ 7. **Find the ratio $a : b$:** $$a : b = a : 3a = 1 : 3$$ **Final answer:** $\boxed{1 : 3}$