It seems like you're asking to solve the quadratic equation:

$4x^2 + 21x + 20 = 0$

We can solve this by factoring, completing the square, or using the quadratic formula. Let's attempt factoring first.

Step 1: Factor the quadratic expression

We need to find two numbers that multiply to $4 \times 20 = 80$ and add up to 21.

After analyzing, the numbers that fit are 16 and 5 because:

$16 \times 5 = 80 \quad \text{and} \quad 16 + 5 = 21$

Step 2: Rewrite the middle term

Now, rewrite the equation:

$4x^2 + 16x + 5x + 20 = 0$

Step 3: Group the terms

Group the terms into pairs:

$(4x^2 + 16x) + (5x + 20) = 0$

Step 4: Factor each group

Factor out the greatest common factors from each group:

$4x(x + 4) + 5(x + 4) = 0$

Step 5: Factor the common binomial

Now, factor out the common binomial $(x + 4)$:

$(4x + 5)(x + 4) = 0$

Step 6: Solve for $x$

Set each factor equal to 0:

$4x + 5 = 0 \quad \text{or} \quad x + 4 = 0$

For $4x + 5 = 0$:

$4x = -5 \quad \Rightarrow \quad x = -\frac{5}{4}$

For $x + 4 = 0$:

$x = -4$

Final Answer:

The solutions are:

$x = -\frac{5}{4} \quad \text{or} \quad x = -4$

Would you like further details or have any questions on this solution?

Here are some related questions:

1. What happens when we use the quadratic formula to solve the equation?
2. Can you factor a quadratic equation if the leading coefficient is 1?
3. What does the discriminant in the quadratic formula represent?
4. How can completing the square help in solving quadratics?
5. What are the conditions for a quadratic equation to be factorable?

Tip: Always check your factors by expanding them back to make sure they match the original quadratic equation.