Factor a^4b^2-6a^2bc+9c^2

$a^{4} b^{2} - 6 a^{2} b c + 9 c^{2}$

Rewrite $a^{4} b^{2}$ as ${(a^{2} b)}^{2}$ .

${(a^{2} b)}^{2} - 6 a^{2} b c + 9 c^{2}$

Rewrite $9 c^{2}$ as ${(3 c)}^{2}$ .

${(a^{2} b)}^{2} - 6 a^{2} b c + {(3 c)}^{2}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 a^{2} b c = 2 \cdot (a^{2} b) \cdot (3 c)$

Rewrite the polynomial.

${(a^{2} b)}^{2} - 2 \cdot (a^{2} b) \cdot (3 c) + {(3 c)}^{2}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = a^{2} b$ and $b = 3 c$ .

${(a^{2} b - 3 c)}^{2}$