Answer
$(3x + 8)^2$
Work Step by Step
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 $9x^2 + 48x + 64$, we look for the factors that, when multiplied together, will equal $(a)(c)$, which is $(9)(64)$ or $576$, but when adding the factors together will equal $b$ or $48$. We need for both factors to be positive because positive numbers, when multiplied together, equal a positive number, and when added together, will make a positive number as well:
$576=(72)(8)$
$72+8 = 80$
$576=(24)(24)$
$24+24 = 48$
$576=(48)(12)$
$48+12 = 60$
The second pair, $24$ and$24$, is what we are looking for.
Split the middle term using this pair of factors as coefficients of $x$ to obtain::
$9x^2 + 24x + 24x + 64$
Now, use factoring by grouping.
Group the first two terms together and the last two terms together:
$(9x^2 + 24x) + (24x + 64)$
$3x$ is a common factor for the first group, and $8$ is a common factor for the second group, so factor them out:
$3x(3x + 8) + 8(3x + 8)$
$3x + 8$ is common to both groups, so factor it out to obtain.
$(3x + 8)(3x + 8)$
Since both factors are the same, we can express it as a single binomial raised to the second power:
$(3x + 8)^2$
