Question

{21x^2-x-10}/{3x+2}

Answer 1

To simplify the expression \(\frac{21x^2 - x - 10}{3x + 2}\), we first need to factor the numerator, \(21x^2 - x - 10\).

To factor \(21x^2 - x - 10\), we can look for two numbers that multiply to \(21 \times -10 = -210\) and add to \(-1\) (the coefficient of \(x\)).

The two numbers that satisfy this condition are \(14\) and \(-15\). We can rewrite the middle term \(-x\) using these numbers:

\[
21x^2 + 14x - 15x - 10
\]

Now, we can group the terms:

\[
(21x^2 + 14x) + (-15x - 10)
\]

Factoring out the common factors in each group gives:

\[
7x(3x + 2) - 5(3x + 2)
\]

Now, we can factor out the common factor \((3x + 2)\):

\[
(7x - 5)(3x + 2)
\]

Now we substitute this factored form back into the original expression:

\[
\frac{(7x - 5)(3x + 2)}{3x + 2}
\]

We can now cancel out the common factor \((3x + 2)\), provided \(3x + 2 \neq 0\) (or equivalently, \(x \neq -\frac{2}{3}\)):

\[
7x - 5
\]

Thus, the simplified form of the expression \(\frac{21x^2 - x - 10}{3x + 2}\) is:

\[
7x - 5, \quad \text{for } x \neq -\frac{2}{3}.
\]