Factor 4a^2+b^2-c^2+4ab

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

Step 1

Regroup terms.

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

Step 2

Factor using the perfect square rule.

Rearrange terms.

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

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

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

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

$4 a b = 2 \cdot (2 a) \cdot b$

Rewrite the polynomial.

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

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

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

Step 3

Reorder $- c^{2}$ and ${(2 a + b)}^{2}$ .

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

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 a + b$ and $b = c$ .

$(2 a + b + c) (2 a + b - c)$

Factor 4a^2+b^2-c^2+4ab

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