Let the unknown number be represented by \( x \). We can translate the problem statement into an equation.
The first part of the statement is "four times the sum of a number and four", which can be expressed as:
\[ 4(x + 4) \]
The second part is "two multiplied by two less than three times the number", which can be expressed as:
\[ 2(3x - 2) \]
Now we can set the two expressions equal to each other, creating the equation:
\[ 4(x + 4) = 2(3x - 2) \]
Next, we will simplify both sides of the equation.
Starting with the left-hand side:
\[ 4(x + 4) = 4x + 16 \]
Now simplifying the right-hand side:
\[ 2(3x - 2) = 6x - 4 \]
Now we have the equation:
\[ 4x + 16 = 6x - 4 \]
To solve for \( x \), we will first move the \( 6x \) term to the left side by subtracting \( 6x \) from both sides:
\[ 4x - 6x + 16 = -4 \] \[ -2x + 16 = -4 \]
Next, we will isolate the \( -2x \) term by subtracting 16 from both sides:
\[ -2x = -4 - 16 \] \[ -2x = -20 \]
Now, divide both sides by -2:
\[ x = \frac{-20}{-2} \] \[ x = 10 \]
Thus, the solution to the problem is:
\[ \boxed{10} \]