1. **Problem statement:** We want to complete the square for the quadratic expression $$4x^2 + 12x + 10$$. 2. **Recall the formula for completing the square:** For any quadratic expression $$ax^2 + bx + c$$, we can rewrite it as $$a(x + d)^2 + e$$ where $$d$$ and $$e$$ are constants found by manipulating the expression. 3. **Step-by-step process:** - Start with the expression: $$4x^2 + 12x + 10$$ - Factor out the coefficient of $$x^2$$ from the first two terms: $$4(x^2 + 3x) + 10$$ - To complete the square inside the parentheses, take half of the coefficient of $$x$$, which is $$3$$, so half is $$\frac{3}{2}$$, and square it: $$\left(\frac{3}{2}\right)^2 = \frac{9}{4}$$ - Add and subtract this inside the parentheses to keep the expression equivalent: $$4\left(x^2 + 3x + \frac{9}{4} - \frac{9}{4}\right) + 10$$ - Group the perfect square trinomial and the constant terms: $$4\left(\left(x + \frac{3}{2}\right)^2 - \frac{9}{4}\right) + 10$$ - Distribute the 4: $$4\left(x + \frac{3}{2}\right)^2 - 4 \times \frac{9}{4} + 10 = 4\left(x + \frac{3}{2}\right)^2 - 9 + 10$$ - Simplify the constants: $$4\left(x + \frac{3}{2}\right)^2 + 1$$ 4. **Final completed square form:** $$4\left(x + \frac{3}{2}\right)^2 + 1$$ 5. **Explanation:** Completing the square rewrites the quadratic in a form that clearly shows its vertex and makes it easier to analyze or integrate. Here, the expression is a perfect square plus a constant, which is useful for solving equations or integrating functions involving this quadratic. 6. **Additional simplification:** Recognize that $$4 = 2^2$$, so: $$4\left(x + \frac{3}{2}\right)^2 + 1 = (2(x + \frac{3}{2}))^2 + 1 = (2x + 3)^2 + 1$$ This matches the expression in the notes. --- **Summary:** $$4x^2 + 12x + 10 = (2x + 3)^2 + 1$$ This is the completed square form of the quadratic expression.