Step 1a

Clear the fractions by finding a common denominator.

In this case, the common denominator for the three fractions is \$x(x+1)(x+2)\$.

Multiply every term by this the common denominator to eliminate the fractions:

\$x(x + 1)(x + 2) ( 3/x + 2/(x + 1) ) = x(x + 1)(x + 2)( 5/(x + 2 )) \$

After simplifying, the equation becomes:

\$ 3(x + 1)(x + 2) + 2x(x + 2) = 5x(x + 1) \$.

Explanation: Here, we multiply the common denominator for the three fractions, and then after simplification, we get \$ 3(x + 1)(x + 2)+ 2x(x + 2) = 5x(x + 1) \$.

Step 1b

Expand and simplify the equation.

\$ 3(x^2 + 3x + 2) + 2(x^2 + 2x) = 5x^2 + 5x \$

Expanding and simplifying further, we get:

\$ 3x^2 + 9x + 6 + 2x^2+ 4x = 5x^2 + 5x \$

Combining like terms:

\$ 5x^2 + 13x + 6 = 5x^2 + 5x \$

Explanation: Here, we simplify the equation then we get \$ 5x^2 + 13x + 6 = 5x^2 + 5x \$.

Step 1c

Move all terms to one side of the equation.

\$ \cancel(5x^2) + 13x + 6 - \cancel(5x^2) - 5x = 0 \$

Simplifying:

\$ 8x + 6 = 0 \$

Explanation: Here, after we simplify the equation then we get a linear equation is \$ 8x + 6 = 0 \$.

Step 1d

Solve the resulting linear equation.

By subtracting 6 from both sides:

\$ 8x + \cancel6 - \cancel6 = 0 - 6 \$

After cancellation :

\$ 8x = - 6 \$

Dividing by 8:

\$ (\cancel(8)^1x)/\cancel8^1 = - \cancel6^3/\cancel8^4 \$

After cancellation:

\$ x = - 3/4 \$

Therefore, the solution to the quadratic equation with rational expression is

\$ x = - 3/4 \$.

Explanation: Therefore, the solutions to the quadratic equation is \$x = - 3/4 \$.