To determine what must be added to each expression to make it a perfect square, we complete the square for each quadratic expression. The general process is to take half the coefficient of $x$, square it, and add the result. Let us solve each part step by step.

Part (a): $x^2 + 8x$

1. Take the coefficient of $x$: $8$.
2. Divide it by $2$: $\frac{8}{2} = 4$.
3. Square the result: $4^2 = 16$.

Thus, 16 must be added to make $x^2 + 8x$ a perfect square: $x^2 + 8x + 16 = (x + 4)^2.$

Part (b): $x^2 + 11x$

1. Take the coefficient of $x$: $11$.
2. Divide it by $2$: $\frac{11}{2} = 5.5$.
3. Square the result: $(5.5)^2 = 30.25$.

Thus, 30.25 must be added to make $x^2 + 11x$ a perfect square: $x^2 + 11x + 30.25 = \left(x + 5.5\right)^2.$

Part (c): $x^2 - 4x$

1. Take the coefficient of $x$: $-4$.
2. Divide it by $2$: $\frac{-4}{2} = -2$.
3. Square the result: $(-2)^2 = 4$.

Thus, 4 must be added to make $x^2 - 4x$ a perfect square: $x^2 - 4x + 4 = (x - 2)^2.$

Final Results:

• (a) Add 16.
• (b) Add 30.25.
• (c) Add 4.

Let me know if you'd like further explanation or a breakdown of the steps!

Here are some related questions for practice:

1. What is the general formula for completing the square for $x^2 + bx$?
2. How do you rewrite a perfect square trinomial as a squared binomial?
3. Can this method be extended to quadratic equations with a coefficient of $x^2$ other than 1?
4. How does completing the square help in solving quadratic equations?
5. What happens if the quadratic term $x^2$ has a negative coefficient?

Tip: Always verify your completed square by expanding the resulting binomial to ensure it matches the original expression!