To solve the equation \((5 + 2x)(2x + 7) = 4x^2 - 25\), let's first expand the left-hand side and then simplify the equation.

1. Expand the left-hand side:
\[
(5 + 2x)(2x + 7)
\]

Using the distributive property (FOIL method):
\[
= 5 \cdot 2x + 5 \cdot 7 + 2x \cdot 2x + 2x \cdot 7
\]
\[
= 10x + 35 + 4x^2 + 14x
\]

Combine like terms:
\[
= 4x^2 + 24x + 35
\]

So the equation becomes:
\[
4x^2 + 24x + 35 = 4x^2 - 25
\]

2. Subtract \(4x^2\) from both sides to simplify:
\[
24x + 35 = -25
\]

3. Subtract 35 from both sides:
\[
24x = -25 - 35
\]
\[
24x = -60
\]

4. Divide both sides by 24:
\[
x = \frac{-60}{24}
\]

Simplify the fraction:
\[
x = \frac{-5}{2}
\]

So, the solution to the equation \((5 + 2x)(2x + 7) = 4x^2 - 25\) is:
\[
x = -\frac{5}{2}
\]