Factor 8x^3+12x^2y+6xy^2+y^3

$8 x^{3} + 12 x^{2} y + 6 x y^{2} + y^{3}$

Regroup terms.

$8 x^{3} + y^{3} + 12 x^{2} y + 6 x y^{2}$

Rewrite $8 x^{3}$ as ${(2 x)}^{3}$ .

${(2 x)}^{3} + y^{3} + 12 x^{2} y + 6 x y^{2}$

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 2 x$ and $b = y$ .

$(2 x + y) ({(2 x)}^{2} - (2 x) y + y^{2}) + 12 x^{2} y + 6 x y^{2}$

Simplify.

Apply the product rule to $2 x$ .

$(2 x + y) (2^{2} x^{2} - (2 x) y + y^{2}) + 12 x^{2} y + 6 x y^{2}$

Raise $2$ to the power of $2$ .

$(2 x + y) (4 x^{2} - (2 x) y + y^{2}) + 12 x^{2} y + 6 x y^{2}$

Multiply $2$ by $- 1$ .

$(2 x + y) (4 x^{2} - 2 x y + y^{2}) + 12 x^{2} y + 6 x y^{2}$

Factor $6 x y$ out of $12 x^{2} y + 6 x y^{2}$ .

Factor $6 x y$ out of $12 x^{2} y$ .

$(2 x + y) (4 x^{2} - 2 x y + y^{2}) + 6 x y (2 x) + 6 x y^{2}$

Factor $6 x y$ out of $6 x y^{2}$ .

$(2 x + y) (4 x^{2} - 2 x y + y^{2}) + 6 x y (2 x) + 6 x y (y)$

Factor $6 x y$ out of $6 x y (2 x) + 6 x y (y)$ .

$(2 x + y) (4 x^{2} - 2 x y + y^{2}) + 6 x y (2 x + y)$

Factor $2 x + y$ out of $(2 x + y) (4 x^{2} - 2 x y + y^{2}) + 6 x y (2 x + y)$ .

Factor $2 x + y$ out of $6 x y (2 x + y)$ .

$(2 x + y) (4 x^{2} - 2 x y + y^{2}) + (2 x + y) (6 x y)$

Factor $2 x + y$ out of $(2 x + y) (4 x^{2} - 2 x y + y^{2}) + (2 x + y) (6 x y)$ .

$(2 x + y) (4 x^{2} - 2 x y + y^{2} + 6 x y)$

Add $- 2 x y$ and $6 x y$ .

$(2 x + y) (4 x^{2} + y^{2} + 4 x y)$

Factor using the perfect square rule.

Rearrange terms.

$(2 x + y) (4 x^{2} + 4 x y + y^{2})$

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$(2 x + y) ({(2 x)}^{2} + 4 x y + y^{2})$

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

$4 x y = 2 \cdot (2 x) \cdot y$

Rewrite the polynomial.

$(2 x + y) ({(2 x)}^{2} + 2 \cdot (2 x) \cdot y + y^{2})$

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

$(2 x + y) {(2 x + y)}^{2}$

Factor 8x^3+12x^2y+6xy^2+y^3

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