Answer
$(4x - 5)^2$
Work Step by Step
First, let's rewrite this expression so that the term with the greatest degree is first:
$16x^2 - 40x + 25$
To factor a quadratic polynomial in the form $ax^2 + bx + c$, we look at factors of $(a)(c)$ such that, when added together, equal the $b$ term.
For the expression $16x^2 - 40x + 25$, we look for the factors that, when multiplied together, will equal $(a)(c)$, which is $(16)(25)$ or $400$, but when adding the factors together will equal $b$ or $-40$. We need for both factors to be negative because negative numbers, when multiplied together, equal a positive number, but when added together, will make a negative number:
$(a)(c)$ = $(-200)(-1)$
$b = -202$
$(a)(c)$ = $(-20)(-20)$
$b = -40$
$(a)(c)$ = $(-40)(-5)$
$b = -45$
The second pair works. We will use that pair to split the middle term:
$16x^2 - 20x - 20x + 25$
Now, we can factor by grouping. We group the first two terms together and the second two terms together:
$(16x^2 - 20x) + (-20x + 25)$
We see that $4x$ is a common factor for the first group, and $-5$ is a common factor for the second group, so let's factor those out:
$4x(4x - 5) - 5(4x - 5)$
We see that $4x - 5$ is common to both groups, so we put that binomial in parentheses. The other binomial will be $4x - 5$, which is composed of the coefficients in front of the binomials. We now have the two factors:
$(4x - 5)(4x - 5)$
Since both factors are the same, we can express it as a single binomial raised to the second power:
$(4x - 5)^2$
