How Do You Solve These Problems?

Parentheses help.
Expression 1 looks like
(3x²) + (x/2)(x)(4x²/3)(x) + 1
Multiplying out the middle term is just like multiplying any sort of fraction: The numerator of the result is the product of the numerators, and the denominator of the result is the product of the denominators. Factors that appear in both the numerator and the denominator cancel each other (because they represent multiplying by 1, which changes nothing).

3x² + (x*4*x²*x)/(2*3) + 1 = 3x² + (x*x²*x)((2*2)/(2*3)) + 1
3x² + 2x⁴/3 + 1  (recognizing that (2*2)/(2*3) = 2/3, and x*x*x² = x⁴)

We can put this all over one denominator:
(3x² + 1)(3/3) + (2x⁴)/3   (grouping, multiplying by 1 terms that don't have 3 as a denominator)
(3(3x²+1) + 2x⁴)/3  (now all terms have 3 as a denominator, so we can combine them over that denominator)

(2x⁴ + 9x² + 3)/3   (multiplying out and putting terms in order by decreasing powers of x)
Please note that if parentheses had been used in the original expression, the "simplification" might be different.