Factor a^8+36b^2c^2+12a^4bc

$a^{8} + 36 b^{2} c^{2} + 12 a^{4} b c$

Rearrange terms.

$a^{8} + 12 a^{4} b c + 36 b^{2} c^{2}$

Rewrite $a^{8}$ as ${(a^{4})}^{2}$ .

${(a^{4})}^{2} + 12 a^{4} b c + 36 b^{2} c^{2}$

Rewrite $36 b^{2} c^{2}$ as ${(6 b c)}^{2}$ .

${(a^{4})}^{2} + 12 a^{4} b c + {(6 b c)}^{2}$

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

$12 a^{4} b c = 2 \cdot a^{4} \cdot (6 b c)$

Rewrite the polynomial.

${(a^{4})}^{2} + 2 \cdot a^{4} \cdot (6 b c) + {(6 b c)}^{2}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a^{4}$ and $b = 6 b c$ .

${(a^{4} + 6 b c)}^{2}$