Start by factoring the coefficient of the squared term into all the terms that have x in its literal part
[tex]\begin{gathered} f(x)=4x^2-16x+13 \\ f(x)=4(x^2-4x)+13 \end{gathered}[/tex]Identify the number that accompanies the x, which in this case is -4, and divide by 2
[tex]-\frac{4}{2}=-2[/tex]then square this term
[tex](-2)^2=4[/tex]now add the last into the parentheses of the factored expression
[tex]f(x)=4(x^2-4x+4)+13[/tex]then since we are adding it to the full equation we also need to substract it in order to mantain the equation the same, but since this is inside the parentheses it is also mutiplied by 4, so the value we are really adding is
[tex](4\cdot4)=16[/tex]Then we also need to substract this value
[tex]f(x)=4(x^2-4x+4)+13-16[/tex]Then the expression that is inside the parentheses is a binomial square we can rewrite as
[tex]f(x)=4\cdot(x-2)^2-3[/tex]we know that the number must be -2 and not +2 because there is a negative sign in the trinomial.
Take into account that
[tex](x\pm a)^2=x^2\pm2\cdot x\cdot a+a^2[/tex]