Simplify and factor the quadratic expression: 3x2 + 7x - 12x - 34 - 2x2 + 10

First, group the terms with the same variable and exponent together.
The given expression is:

$$3x^2 + 7x - 12x - 34 - 2x^2 + 10$$

Group the $x^2$ terms, $x$ terms, and constant terms:

$$(3x^2 - 2x^2) + (7x - 12x) + (-34 + 10)$$

Perform the addition and subtraction for each group of like terms:

$$(3-2)x^2 + (7-12)x + (-34+10)$$ $$1x^2 - 5x - 24$$

The simplified expression is $x^2 - 5x - 24$.

For the quadratic expression $x^2 - 5x - 24$, we need to find two numbers that:

• Multiply to $-24$ (the constant term).
• Add up to $-5$ (the coefficient of the $x$ term). Let's consider factors of $-24$: $1 \times -24 = -24$, $1 + (-24) = -23$ $2 \times -12 = -24$, $2 + (-12) = -10$ $3 \times -8 = -24$, $3 + (-8) = -5$ The two numbers are 3 and $-8$.

Using the two numbers found in Step 3, the quadratic expression $x^2 - 5x - 24$ can be factored as $(x+p)(x+q)$, where $p$ and $q$ are the numbers found.

$$(x+3)(x-8)$$