Let's break down the progression of the equation step by step from the image you uploaded:

The initial expression is:

$-3(1 + \sin^2x) + 2(1 + \cos^2x)$

Step 1: Expand the terms inside the parentheses

Apply the distributive property to both terms:

$-3(1 + \sin^2x) = -3 - 3\sin^2x$ $2(1 + \cos^2x) = 2 + 2\cos^2x$

Now combine them:

$-3 - 3\sin^2x + 2 + 2\cos^2x$

Step 2: Simplify the constants

Combine $-3$ and $2$:

$(-3 + 2) = -1$

So, the expression becomes:

$-1 - 3\sin^2x + 2\cos^2x$

Step 3: Use a trigonometric identity

We know that $\cos^2x = 1 - \sin^2x$ from the Pythagorean identity:

$\cos^2x = 1 - \sin^2x$

Substitute this into the equation:

$-1 - 3\sin^2x + 2(1 - \sin^2x)$

Step 4: Expand the second part

Now distribute the $2$ over $(1 - \sin^2x)$:

$2(1 - \sin^2x) = 2 - 2\sin^2x$

Substitute this back into the equation:

$-1 - 3\sin^2x + 2 - 2\sin^2x$

Step 5: Simplify further

Combine the constants and the $\sin^2x$ terms:

$(-1 + 2) = 1$

$-3\sin^2x - 2\sin^2x = -5\sin^2x$

Thus, the final simplified expression is:

$1 - 5\sin^2x$

This is the result after all simplifications.

Would you like further details or explanations about any specific step?

Here are some follow-up questions to deepen your understanding:

1. What is the significance of the Pythagorean identity in trigonometric simplifications?
2. How does distributing constants across parentheses affect trigonometric terms?
3. Why is combining like terms important in algebraic and trigonometric expressions?
4. Could you expand other similar trigonometric expressions using the same method?
5. How can you use other trigonometric identities to simplify different expressions?

Tip: Always look for known identities like the Pythagorean identity to simplify trigonometric expressions efficiently.