Final answer:
The expression E = (5x + 4y)^3 + (10x + 8y)^2 + 5x + 12y can be factorized as (15x + 12y)(75x^2 + 120xy + 48y^2) + 5x + 12y. The correct answer is d) (5x + 4y)^3 + (10x + 8y)(5x + 4y) + 5x + 12y.
Step-by-step explanation:
The given expression is: E = (5x + 4y)^3 + (10x + 8y)^2 + 5x + 12y
To factorize this expression, we can use the formula for a^3 + b^3 which is (a + b)(a^2 - ab + b^2). Applying this formula, we can rewrite the expression as:
E = [(5x + 4y) + (10x + 8y)][(5x + 4y)^2 - (5x + 4y)(10x + 8y) + (10x + 8y)^2] + 5x + 12y
Simplifying further, we get:
E = (5x + 4y + 10x + 8y)[(5x + 4y)^2 - (5x + 4y)(10x + 8y) + (10x + 8y)^2] + 5x + 12y
Expanding the square terms, we have:
E = (15x + 12y)[25x^2 + 40xy + 16y^2 - 50x^2 - 80xy - 32y^2 + 100x^2 + 160xy + 64y^2] + 5x + 12y
Simplifying further, we get:
E = (15x + 12y)(75x^2 + 120xy + 48y^2) + 5x + 12y
So, the factorized form of E is (15x + 12y)(75x^2 + 120xy + 48y^2) + 5x + 12y.
Therefore, the correct answer is d) (5x + 4y)^3 + (10x + 8y)(5x + 4y) + 5x + 12y.