Express \( 4 x^{2}+20 x+5 \) in the form \( (a x+b)^{2}+c \), where \( a, b \) and \( c \) are constants to be found.

To express \( 4x^{2} + 20x + 5 \) in the form \( (ax + b)^{2} + c \), we can start by completing the square for the quadratic part. 1. **Factor out the coefficient of \( x^2 \)** (which is 4) from the first two terms: \[ 4(x^{2} + 5x) + 5 \] 2. **Complete the square for \( x^{2} + 5x \)**: - Take half of the coefficient of \( x \) (which is 5), square it, and add/subtract it inside the parentheses. - Half of 5 is \( \frac{5}{2} \), and squaring it gives \( \left(\frac{5}{2}\right)^{2} = \frac{25}{4} \). 3. **Incorporate this square into the expression**: \[ 4\left(x^{2} + 5x + \frac{25}{4} - \frac{25}{4}\right) + 5 \] This simplifies to: \[ 4\left(\left(x + \frac{5}{2}\right)^{2} - \frac{25}{4}\right) + 5 \] 4. **Distribute the 4**: \[ 4\left(x + \frac{5}{2}\right)^{2} - 4 \cdot \frac{25}{4} + 5 \] Which simplifies to: \[ 4\left(x + \frac{5}{2}\right)^{2} - 25 + 5 \] Thus: \[ 4\left(x + \frac{5}{2}\right)^{2} - 20 \] 5. **Therefore, we can write the expression in the desired form**: \[ (2x + 5)^{2} - 20 \] In conclusion, we have: \[ a = 2, \quad b = 5, \quad c = -20 \] So the final expression is: \[ \boxed{(2x + 5)^{2} - 20} \]