Answer
$9(k+1)^2$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
(3k+5)^2-4(3k+5)+4
,$ use substitution. Then use the factoring of trinomials in the form $x^2+bx+c.$
$\bf{\text{Solution Details:}}$
Let $z=(3k+5).$ Then the given expression is equivalent to
\begin{array}{l}\require{cancel}
z^2-4z+4
.\end{array}
In the trinomial expression above, $b=
-4
,\text{ and } c=
4
.$ Using the factoring of trinomials in the form $x^2+bx+c,$ the two numbers whose product is $c$ and whose sum is $b$ are $\left\{
-2,-2
\right\}.$ Using these two numbers, the factored form of the expression above is
\begin{array}{l}\require{cancel}
(z-2)(z-2)
\\\\=
(z-2)^2
.\end{array}
Since $z=(3k+5),$ by back substitution, the expression above is equivalent to
\begin{array}{l}\require{cancel}
(3k+5-2)^2
\\\\=
(3k+3)^2
\\\\=
[3(k+1)]^2
\\\\=
3^2(k+1)^2
\\\\=
9(k+1)^2
.\end{array}