Factor x^4-25x^2+3x^3-75x+2x^2-50

$x^{4} - 25 x^{2} + 3 x^{3} - 75 x + 2 x^{2} - 50$

Regroup terms.

$x^{4} + 3 x^{3} + 2 x^{2} - 25 x^{2} - 75 x - 50$

Factor $x^{2}$ out of $x^{4} + 3 x^{3} + 2 x^{2}$ .

Factor $x^{2}$ out of $x^{4}$ .

$x^{2} x^{2} + 3 x^{3} + 2 x^{2} - 25 x^{2} - 75 x - 50$

Factor $x^{2}$ out of $3 x^{3}$ .

$x^{2} x^{2} + x^{2} (3 x) + 2 x^{2} - 25 x^{2} - 75 x - 50$

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

$x^{2} x^{2} + x^{2} (3 x) + x^{2} \cdot 2 - 25 x^{2} - 75 x - 50$

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (3 x)$ .

$x^{2} (x^{2} + 3 x) + x^{2} \cdot 2 - 25 x^{2} - 75 x - 50$

Factor $x^{2}$ out of $x^{2} (x^{2} + 3 x) + x^{2} \cdot 2$ .

$x^{2} (x^{2} + 3 x + 2) - 25 x^{2} - 75 x - 50$

Factor.

Factor $x^{2} + 3 x + 2$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Write the factored form using these integers.

$x^{2} ((x + 1) (x + 2)) - 25 x^{2} - 75 x - 50$

Remove unnecessary parentheses.

$x^{2} (x + 1) (x + 2) - 25 x^{2} - 75 x - 50$

Factor $25$ out of $- 25 x^{2} - 75 x - 50$ .

Factor $25$ out of $- 25 x^{2}$ .

$x^{2} (x + 1) (x + 2) + 25 (- x^{2}) - 75 x - 50$

Factor $25$ out of $- 75 x$ .

$x^{2} (x + 1) (x + 2) + 25 (- x^{2}) + 25 (- 3 x) - 50$

Factor $25$ out of $- 50$ .

$x^{2} (x + 1) (x + 2) + 25 (- x^{2}) + 25 (- 3 x) + 25 (- 2)$

Factor $25$ out of $25 (- x^{2}) + 25 (- 3 x)$ .

$x^{2} (x + 1) (x + 2) + 25 (- x^{2} - 3 x) + 25 (- 2)$

Factor $25$ out of $25 (- x^{2} - 3 x) + 25 (- 2)$ .

$x^{2} (x + 1) (x + 2) + 25 (- x^{2} - 3 x - 2)$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 2 = 2$ and whose sum is $b = - 3$ .

Factor $- 3$ out of $- 3 x$ .

$x^{2} (x + 1) (x + 2) + 25 (- x^{2} - 3 (x) - 2)$

Rewrite $- 3$ as $- 1$ plus $- 2$

$x^{2} (x + 1) (x + 2) + 25 (- x^{2} + (- 1 - 2) x - 2)$

Apply the distributive property.

$x^{2} (x + 1) (x + 2) + 25 (- x^{2} - 1 x - 2 x - 2)$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$x^{2} (x + 1) (x + 2) + 25 ((- x^{2} - 1 x) - 2 x - 2)$

Factor out the greatest common factor (GCF) from each group.

$x^{2} (x + 1) (x + 2) + 25 (x (- x - 1) + 2 (- x - 1))$

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

$x^{2} (x + 1) (x + 2) + 25 ((- x - 1) (x + 2))$

Remove unnecessary parentheses.

$x^{2} (x + 1) (x + 2) + 25 (- x - 1) (x + 2)$

Factor $x + 2$ out of $x^{2} (x + 1) (x + 2) + 25 (- x - 1) (x + 2)$ .

Factor $x + 2$ out of $x^{2} (x + 1) (x + 2)$ .

$(x + 2) (x^{2} (x + 1)) + 25 (- x - 1) (x + 2)$

Factor $x + 2$ out of $25 (- x - 1) (x + 2)$ .

$(x + 2) (x^{2} (x + 1)) + (x + 2) (25 (- x - 1))$

Factor $x + 2$ out of $(x + 2) (x^{2} (x + 1)) + (x + 2) (25 (- x - 1))$ .

$(x + 2) (x^{2} (x + 1) + 25 (- x - 1))$

Apply the distributive property.

$(x + 2) (x^{2} x + x^{2} \cdot 1 + 25 (- x - 1))$

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$(x + 2) (x^{2} x^{1} + x^{2} \cdot 1 + 25 (- x - 1))$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x + 2) (x^{2 + 1} + x^{2} \cdot 1 + 25 (- x - 1))$

Add $2$ and $1$ .

$(x + 2) (x^{3} + x^{2} \cdot 1 + 25 (- x - 1))$

Multiply $x^{2}$ by $1$ .

$(x + 2) (x^{3} + x^{2} + 25 (- x - 1))$

Apply the distributive property.

$(x + 2) (x^{3} + x^{2} + 25 (- x) + 25 \cdot - 1)$

Multiply $- 1$ by $25$ .

$(x + 2) (x^{3} + x^{2} - 25 x + 25 \cdot - 1)$

Multiply $25$ by $- 1$ .

$(x + 2) (x^{3} + x^{2} - 25 x - 25)$

Factor.

Rewrite $x^{3} + x^{2} - 25 x - 25$ in a factored form.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x + 2) ((x^{3} + x^{2}) - 25 x - 25)$

Factor out the greatest common factor (GCF) from each group.

$(x + 2) (x^{2} (x + 1) - 25 (x + 1))$

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$(x + 2) ((x + 1) (x^{2} - 25))$

Rewrite $25$ as $5^{2}$ .

$(x + 2) ((x + 1) (x^{2} - 5^{2}))$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$(x + 2) ((x + 1) ((x + 5) (x - 5)))$

Remove unnecessary parentheses.

$(x + 2) ((x + 1) (x + 5) (x - 5))$

Remove unnecessary parentheses.

$(x + 2) (x + 1) (x + 5) (x - 5)$

Factor x^4-25x^2+3x^3-75x+2x^2-50

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