Factor f² - 16g² + 24gh - 9h² completely.

Step-by-step solution

1. 1

   Look for a hidden perfect square among the last three terms. Isolate them and factor out $-1$:

   $-16g^2+24gh-9h^2=-\left(16g^2-24gh+9h^2\right)$

   Inside the bracket, $16g^2=(4g)^2$, $9h^2=(3h)^2$, and the middle term matches $2(4g)(3h)=24gh$ — so it is a perfect square, $(4g-3h)^2$.

2. 2

   Rewrite the whole expression as a difference of two squares.

   $f^2-16g^2+24gh-9h^2=f^2-(4g-3h)^2$

   The reason to group $3+1$ rather than $2+2$ is precisely that these three terms form a square while $f^2$ stands alone as another.

3. 3

   Apply $a^2-b^2=(a-b)(a+b)$ with $a=f$ and $b=4g-3h$:

   $f^2-(4g-3h)^2=\bigl[f-(4g-3h)\bigr]\bigl[f+(4g-3h)\bigr]$

4. 4

   Remove the inner brackets, minding the signs.

   $f-(4g-3h)=f-4g+3h,\qquad f+(4g-3h)=f+4g-3h$

   Note the first bracket flips both inner signs — the classic slip is writing $f-4g-3h$.

5. 5

   Expand back and test numerically. $(f-4g+3h)(f+4g-3h)=f^2-(4g-3h)^2=f^2-16g^2+24gh-9h^2$ ✓. At $f=5$, $g=1$, $h=1$: the original is $25-16+24-9=24$, and $(5-4+3)(5+4-3)=4\cdot 6=24$ ✓. Neither bracket factors further, so the factoring is complete.