To simplify the expression \(\sqrt{21 + 4\sqrt{5}}\), we can first check if it can be expressed in the form \(\sqrt{a} + \sqrt{b}\).

Let's assume:

\[
\sqrt{21 + 4\sqrt{5}} = \sqrt{a} + \sqrt{b}
\]

Now, squaring both sides:

\[
21 + 4\sqrt{5} = (\sqrt{a} + \sqrt{b})^2
\]

Expanding the right-hand side:

\[
21 + 4\sqrt{5} = a + b + 2\sqrt{ab}
\]

Now, compare the rational and irrational parts from both sides of the equation:

1. **Rational part:** \(a + b = 21\)
2. **Irrational part:** \(2\sqrt{ab} = 4\sqrt{5}\)

From the second equation, we solve for \(ab\):

\[
2\sqrt{ab} = 4\sqrt{5} \quad \Rightarrow \quad \sqrt{ab} = 2\sqrt{5} \quad \Rightarrow \quad ab = 20
\]

Now, we have a system of equations:

\[
a + b = 21
\]
\[
ab = 20
\]

We can solve this system using the quadratic equation. The quadratic equation formed by these two conditions is:

\[
x^2 - (a + b)x + ab = 0
\]

Substituting the values:

\[
x^2 - 21x + 20 = 0
\]

Let's solve this quadratic equation for \(x\).The solutions to the quadratic equation \(x^2 - 21x + 20 = 0\) are \(x = 1\) and \(x = 20\). 

Thus, \(a = 1\) and \(b = 20\). Therefore:

\[
\sqrt{21 + 4\sqrt{5}} = \sqrt{1} + \sqrt{20} = 1 + 2\sqrt{5}
\]

So, the simplified form of \(\sqrt{21 + 4\sqrt{5}}\) is \(1 + 2\sqrt{5}\).

Would you like more details or have any questions about this solution?

Here are some related questions:

1. How do you simplify a square root expression in general?
2. What are the properties of square roots that can be applied in algebraic expressions?
3. How do you solve quadratic equations by factoring?
4. Can all square roots be simplified in the form \(\sqrt{a} + \sqrt{b}\)?
5. What other methods can be used to simplify expressions involving square roots?

**Tip:** When dealing with square root expressions, it's often helpful to assume a form and then verify it by squaring and solving for unknowns.

Learn how to simplify the square root of 21 plus 4 times the square root of 5 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the simplification of radicals and the quadratic equation solution. Suitable for high school students in Grades 10-12.

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