Simplify (a^2-10a+21)/(a-7)*(a^2+a-12)/((a-3)^2)

$\frac{a^{2} - 10 a + 21}{a - 7} \cdot \frac{a^{2} + a - 12}{{(a - 3)}^{2}}$

Step 1

Factor $a^{2} - 10 a + 21$ 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 $21$ and whose sum is $- 10$ .

$- 7, - 3$

Write the factored form using these integers.

$\frac{(a - 7) (a - 3)}{a - 7} \cdot \frac{a^{2} + a - 12}{{(a - 3)}^{2}}$

Step 2

Factor $a^{2} + a - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Write the factored form using these integers.

$\frac{(a - 7) (a - 3)}{a - 7} \cdot \frac{(a - 3) (a + 4)}{{(a - 3)}^{2}}$

Step 3

Simplify terms.

Cancel the common factor of $a - 3$ .

Factor $a - 3$ out of $(a - 7) (a - 3)$ .

$\frac{(a - 3) (a - 7)}{a - 7} \cdot \frac{(a - 3) (a + 4)}{{(a - 3)}^{2}}$

Factor $a - 3$ out of ${(a - 3)}^{2}$ .

$\frac{(a - 3) (a - 7)}{a - 7} \cdot \frac{(a - 3) (a + 4)}{(a - 3) (a - 3)}$

Cancel the common factor.

$\frac{(a - 3) (a - 7)}{a - 7} \cdot \frac{(a - 3) (a + 4)}{(a - 3) (a - 3)}$

Rewrite the expression.

$\frac{a - 7}{a - 7} \cdot \frac{(a - 3) (a + 4)}{a - 3}$

Multiply $\frac{a - 7}{a - 7}$ by $\frac{(a - 3) (a + 4)}{a - 3}$ .

$\frac{(a - 7) ((a - 3) (a + 4))}{(a - 7) (a - 3)}$

Cancel the common factor of $a - 7$ .

Cancel the common factor.

$\frac{(a - 7) ((a - 3) (a + 4))}{(a - 7) (a - 3)}$

Rewrite the expression.

$\frac{(a - 3) (a + 4)}{a - 3}$

Cancel the common factor of $a - 3$ .

Cancel the common factor.

$\frac{(a - 3) (a + 4)}{a - 3}$

Divide $a + 4$ by $1$ .

$a + 4$

Simplify (a^2-10a+21)/(a-7)*(a^2+a-12)/((a-3)^2)

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