Evaluate: f(x)=4x^2+8x-5

Expression: $f\left( x \right)=4{x}^{2}+8x-5$

To find the $x$-intercept/zero, substitute $f\left( x \right)=0$

$0=4{x}^{2}+8x-5$

Swap the sides of the equation

$4{x}^{2}+8x-5=0$

Write $8x$ as a difference

$4{x}^{2}+10x-2x-5=0$

Factor out $2x$ from the expression

$2x \times \left( 2x+5 \right)-2x-5=0$

Factor out the negative sign from the expression

$2x \times \left( 2x+5 \right)-\left( 2x+5 \right)=0$

Factor out $2x+5$ from the expression

$\left( 2x+5 \right) \times \left( 2x-1 \right)=0$

When the product of factors equals $0$, at least one factor is $0$

$\begin{array} { l }2x+5=0,\\2x-1=0\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=-\frac{ 5 }{ 2 },\\2x-1=0\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=-\frac{ 5 }{ 2 },\\x=\frac{ 1 }{ 2 }\end{array}$

The equation has $2$ solutions

$\begin{align*}&\begin{array} { l }x_1=-\frac{ 5 }{ 2 },& x_2=\frac{ 1 }{ 2 }\end{array} \\&\begin{array} { l }x_1=-2.5,& x_2=0.5\end{array}\end{align*}$