Answer
$\left( 2t^{\frac{1}{2}}-5 \right)^2$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
4t-20t^{\frac{1}{2}}+25
,$ find two numbers whose product is $ac$ and whose sum is $b$ in the quadratic expression $ax^2+bx+c.$ Use these $2$ numbers to decompose the middle term of the given quadratic expression and then use factoring by grouping.
$\bf{\text{Solution Details:}}$
Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $
4(25)=100
$ and the value of $b$ is $
-20
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
-10,-10
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
4t-10t^{\frac{1}{2}}-10t^{\frac{1}{2}}+25
.\end{array}
Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to
\begin{array}{l}\require{cancel}
\left( 4t-10t^{\frac{1}{2}} \right)- \left( 10t^{\frac{1}{2}}-25 \right)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
2t^{\frac{1}{2}}\left( 2t^{\frac{1}{2}}-5 \right)- 5\left( 2t^{\frac{1}{2}}-5 \right)
.\end{array}
Factoring the $GCF=
\left( 2t^{\frac{1}{2}}-5 \right)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
\left( 2t^{\frac{1}{2}}-5 \right)\left(2t^{\frac{1}{2}}- 5\right)
\\\\=
\left( 2t^{\frac{1}{2}}-5 \right)^2
.\end{array}
