Let's define the variable as \( x \). The given statement can be translated into a mathematical equation as follows:
- "The product of 4 and a number squared" translates to \( 4x^2 \).
- "The product of 8 and the same number" translates to \( 8x \).
- "is 7" means we will set the expression equal to 7.
Putting it all together, the equation becomes:
\[ 4x^2 - 8x = 7 \]
Now, we can rearrange it to have all terms on one side of the equation:
\[ 4x^2 - 8x - 7 = 0 \]
Therefore, the equivalent quadratic equation is:
\[ 4x^2 - 8x - 7 = 0 \]