1. **State the problem:** We need to express the quadratic expression $4x^2 - 12x + 13$ in the form $(2x + a)^2 + b$, where $a$ and $b$ are constants. 2. **Recall the formula:** The expression $(2x + a)^2$ expands to $4x^2 + 4ax + a^2$. We want to match this to $4x^2 - 12x + 13$ by finding $a$ and $b$ such that: $$4x^2 - 12x + 13 = (2x + a)^2 + b = 4x^2 + 4ax + a^2 + b$$ 3. **Match coefficients:** Comparing the coefficients of $x$ terms: $$-12x = 4ax \implies 4a = -12 \implies a = \frac{-12}{4} = -3$$ 4. **Substitute $a$ back:** Now substitute $a = -3$ into the expression: $$(2x - 3)^2 + b = 4x^2 + 4(-3)x + (-3)^2 + b = 4x^2 - 12x + 9 + b$$ 5. **Find $b$ by matching constants:** The original constant term is 13, so: $$9 + b = 13 \implies b = 13 - 9 = 4$$ 6. **Final expression:** Therefore, $$4x^2 - 12x + 13 = (2x - 3)^2 + 4$$ This completes the problem by expressing the quadratic in the desired form.