Question

Can you explain how to simplify this problem in standard form and show the work please?
(5x+4) (3x+2) + (X^2-6) (X-2) Over (x-2). The (x^2-6) (x-2) is on the top with a line drawn under it with (x-2) below it. it's division I think thank you

Answer 1

To simplify the problem \(\frac{(5x+4)(3x+2)+(x^2-6)(x-2)}{x-2}\), we can follow these steps:

### Step 1: Expand the Numerator

First, we need to expand the terms in the numerator, which consists of two parts: \((5x+4)(3x+2)\) and \((x^2-6)(x-2)\).

**1.** Expand \((5x + 4)(3x + 2)\):

\[
(5x + 4)(3x + 2) = 5x \cdot 3x + 5x \cdot 2 + 4 \cdot 3x + 4 \cdot 2
\]
\[
= 15x^2 + 10x + 12x + 8 = 15x^2 + 22x + 8
\]

**2.** Expand \((x^2 - 6)(x - 2)\):

\[
(x^2 - 6)(x - 2) = x^2 \cdot x + x^2 \cdot (-2) - 6 \cdot x - 6 \cdot (-2)
\]
\[
= x^3 - 2x^2 - 6x + 12
\]

### Step 2: Combine the Expanded Terms

Now, we combine both expanded parts:

\[
15x^2 + 22x + 8 + (x^3 - 2x^2 - 6x + 12)
\]

Combine like terms:
\[
x^3 + (15x^2 - 2x^2) + (22x - 6x) + (8 + 12)
\]
\[
= x^3 + 13x^2 + 16x + 20
\]

Thus, the numerator can be expressed as:
\[
x^3 + 13x^2 + 16x + 20
\]

### Step 3: Put it All Together

Putting everything together, the expression simplifies to:

\[
\frac{x^3 + 13x^2 + 16x + 20}{x - 2}
\]

### Step 4: Perform Polynomial Long Division

Now, we need to perform polynomial long division of \(x^3 + 13x^2 + 16x + 20\) by \(x - 2\).

1. Divide the leading term: \(x^3 \div x = x^2\).
2. Multiply \(x^2\) by \(x - 2\):
\[
x^2(x - 2) = x^3 - 2x^2
\]
3. Subtract:
\[
(x^3 + 13x^2 + 16x + 20) - (x^3 - 2x^2) = 15x^2 + 16x + 20
\]

4. Repeat:
- Divide the leading term: \(15x^2 \div x = 15x\).
- Multiply \(15x\) by \(x - 2\):
\[
15x(x - 2) = 15x^2 - 30x
\]
5. Subtract:
\[
(15x^2 + 16x + 20) - (15x^2 - 30x) = 46x + 20
\]

6. Repeat:
- Divide the leading term: \(46x \div x = 46\).
- Multiply \(46\) by \(x - 2\):
\[
46(x - 2) = 46x - 92
\]
7. Subtract:
\[
(46x + 20) - (46x - 92) = 112
\]

### Result of Division

The result of the polynomial long division is:

\[
x^2 + 15x + 46 + \frac{112}{x - 2}
\]

### Final Answer

Thus, the simplified form of the original expression is:

\[
x^2 + 15x + 46 + \frac{112}{x - 2}
\]

This is the simplified expression in standard form.