To solve the quadratic equation 4x^2 + 5x + 1 = 0 using the cross-multiplication method, we need to find two numbers whose product is equal to 4 * 1 = 4 (coefficient of x^2 times the constant term) and whose sum is equal to the coefficient of x (5).
The numbers that satisfy these conditions are 4 and 1.
Next, we rewrite the quadratic equation as:
4x^2 + 4x + x + 1 = 0
Now we group the terms together:
(4x^2 + 4x) + (x + 1) = 0
Next, we factor out the common factors from each group:
4x(x + 1) + 1(x + 1) = 0
Now, we can see that we have a common factor of (x + 1):
(x + 1)(4x + 1) = 0
Now, we set each factor equal to zero and solve for x:
x + 1 = 0 or 4x + 1 = 0
Solving the first equation, x + 1 = 0, gives x = -1.
Solving the second equation, 4x + 1 = 0, gives x = -1/4.
Therefore, the solutions to the quadratic equation are x = -1 and x = -1/4.
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